First, The TLS handshake is the Roku disconnecting / losing connection (reset by peer). The Roku is disconnecting for some reason until I look further down and see where your connection is dropping BUT also your database is showing signs of being very fragmented as I outlined above.
Jul 01, 2017 16:35:07.640 [0x7fcaae3ff700] DEBUG - Request: [127.0.0.1:54224 (Loopback)] PUT /video/:/transcode/session/2d2c4f20e6311d80321e06914291cb28/386fde02-9d0e-4f20-9a74-cd935a5210eb/progress?progress=74.8&size=-22&remaining=-1&vdec_packets=93302&vdec_sw_ok=93296&speed=0.0&vdec_hw_status=0 (18 live) Signed-in Token (korzynski)
Jul 01, 2017 16:35:07.640 [0x7fcad2fff700] ERROR - Exception getting remote address: remote_endpoint: Transport endpoint is not connected
Jul 01, 2017 16:35:07.642 [0x7fcad27fe700] ERROR - CERT: Error in TLS handshake: Connection reset by peer
Jul 01, 2017 16:35:07.642 [0x7fcad2fff700] DEBUG - Auth: We found auth token (xxxxxxxxxxxxxxxxxxxx), enabling token-based authentication.
Jul 01, 2017 16:35:07.642 [0x7fcad2fff700] DEBUG - Auth: Came in with the master token, authorization succeeded.
Jul 01, 2017 16:39:02.894 [0x7fcad2fff700] DEBUG - EventSource: Failure in IdleTimeout (0 - Success).
Jul 01, 2017 16:39:02.894 [0x7fcad2fff700] DEBUG - MyPlex: We appear to have lost Internet connectivity, resetting device URL cache.
Jul 01, 2017 16:39:02.894 [0x7fcad2fff700] ERROR - EventSource: Retrying in 15 seconds.
Jul 01, 2017 16:39:02.895 [0x7fcaa53f8700] DEBUG - It took 90.000000 ms to retrieve 1 items.
Lastly, here is your system, sending Sernity to the remote Roku. The activity
progress=74.9&size=-22&remaining=-1&vdec_packets=93387&vdec_sw_ok=93381&speed=0.0&vdec_hw_status=0 (36 live) 0ms 342 bytes
Jul 01, 2017 16:39:02.907 [0x7fcaa8bff700] DEBUG - [Transcoder] Throttle - Going into sloth mode.
Jul 01, 2017 16:39:02.942 [0x7fcaa63fa700] DEBUG - Play progress on 17228 ‘Serenity’ - got played 3650000 ms by account 728504!
Jul 01, 2017 16:39:02.943 [0x7fcac6bfe700] DEBUG - Statistics: (2d2c4f20e6311d80321e06914291cb28) Reporting active playback in state 0 of type 4 (scrobble: 0) for account 728504
Jul 01, 2017 16:39:02.943 [0x7fcaa5bf9700] DEBUG - Client [2d2c4f20e6311d80321e06914291cb28] reporting timeline state playing, progress of 3682000/5202198ms for guid=com.plexapp.agents.thetvdbdvdorder://78874/1/1?lang=en, ratingKey=17228 url=, key=/library/metadata/17228, containerKey=, metadataId=17228
Jul 01, 2017 16:39:02.944 [0x7fcac93ff700] DEBUG - [Now] User is terirhan (ID: 728504)
Jul 01, 2017 16:39:02.944 [0x7fcac93ff700] DEBUG - [Now] Device is Roku (Roku 1 - 1XJ56A020198).
Jul 01, 2017 16:39:02.944 [0x7fcac93ff700] DEBUG - [Now] Profile is Roku-7.x
Jul 01, 2017 16:39:02.944 [0x7fcad2fff700] DEBUG - Completed: [10.55.55.1:52701] 200 GET /:/timeline?time=3492000&duration=5202198&key=%2Flibrary%2Fmetadata%2F17228&ratingKey=17228&state=playing&guid=com.plexapp.agents.thetvdbdvdorder%3A%2F%2F78874%2F1%2F1%3Flang%3Den&playQueueItemID=62892 (36 live) TLS GZIP 175411ms 710 bytes (pipelined: 1)
Jul 01, 2017 16:39:02.944 [0x7fcac93ff700] DEBUG - [Now] Updated play state for /library/metadata/17228.
Jul 01, 2017 16:39:02.945 [0x7fcaad3fd700] DEBUG - Play progress on 17228 ‘Serenity’ - got played 3671000 ms by account 728504!
Jul 01, 2017 16:39:02.971 [0x7fcac13f3700] DEBUG - Request: [10.55.55.1:51314 (Subnet)] GET /:/timeline?time=3692000&duration=5202198&key=%2Flibrary%2Fmetadata%2F17228&ratingKey=17228&state=playing&guid=com.plexapp.agents.thetvdbdvdorder%3A%2F%2F78874%2F1%2F1%3Flang%3Den&playQueueItemID=62892 (35 live) TLS GZIP Signed-in Token (terirhan)
Jul 01, 2017 16:39:02.971 [0x7fcad2fff700] DEBUG - Auth: We found auth token (xxxxxxxxxxxxxxxxxxxx), enabling token-based authentication.
Jul 01, 2017 16:39:02.971 [0x7fcad2fff700] DEBUG - Auth: Came in with the master token, authorization succeeded.
Jul 01, 2017 16:39:02.972 [0x7fcaa6bfb700] DEBUG - Request: [127.0.0.1:54706 (Loopback)] PUT /video/:/transcode/session/2d2c4f20e6311d80321e06914291cb28/386fde02-9d0e-4f20-9a74-cd935a5210eb/progress?progress=74.9&size=-22&remaining=-1&vdec_packets=93382&vdec_sw_ok=93376&speed=0.0&vdec_hw_status=0 (34 live) Signed-in Token (korzynski)
Jul 01, 2017 16:39:02.972 [0x7fcad27fe700] DEBUG - Completed: [127.0.0.1:54706] 206 PUT /video/:/transcode/session/2d2c4f20e6311d80321e06914291cb28/386fde02-9d0e-4f20-9a74-cd935a5210eb/progress?progress=74.9&size=-22&remaining=-1&vdec_packets=93382&vdec_sw_ok=93376&speed=0.0&vdec_hw_status=0 (34 live) 0ms 342 bytes
Jul 01, 2017 16:39:03.000 [0x7fcad2fff700] DEBUG - Auth: We found auth token (xxxxxxxxxxxxxxxxxxxx), enabling token-based authentication.
Jul 01, 2017 16:39:03.000 [0x7fcad2fff700] DEBUG - Auth: authenticated user 728504 as tn
Jul 01, 2017 16:39:03.017 [0x7fcac13f3700] DEBUG - Client [2d2c4f20e6311d80321e06914291cb28] reporting timeline state playing, progress of 3692000/5202198ms for guid=com.plexapp.agents.thetvdbdvdorder://78874/1/1?lang=en, ratingKey=17228 url=, key=/library/metadata/17228, containerKey=, metadataId=17228
Jul 01, 2017 16:39:03.018 [0x7fcad27fe700] DEBUG - Completed after connection close: [10.55.55.1:52704] 200 GET /:/timeline?time=3513000&duration=5202198&key=%2Flibrary%2Fmetadata%2F17228&ratingKey=17228&state=playing&guid=com.plexapp.agents.thetvdbdvdorder%3A%2F%2F78874%2F1%2F1%3Flang%3Den&playQueueItemID=62892 (34 live) TLS GZIP 175483ms 0 bytes (pipelined: 2)
Jul 01, 2017 16:39:03.020 [0x7fcaa2bf3700] DEBUG - Request: [10.55.55.1:51318 (Subnet)] GET /:/timeline?time=3713000&duration=5202198&key=%2Flibrary%2Fmetadata%2F17228&ratingKey=17228&state=playing&guid=com.plexapp.agents.thetvdbdvdorder%3A%2F%2F78874%2F1%2F1%3Flang%3Den&playQueueItemID=62892 (34 live) TLS GZIP Signed-in Token (terirhan)
Jul 01, 2017 16:39:03.021 [0x7fcaa2bf3700] DEBUG - Client [2d2c4f20e6311d80321e06914291cb28] reporting timeline state playing, progress of 3713000/5202198ms for guid=com.plexapp.agents.thetvdbdvdorder://78874/1/1?lang=en, ratingKey=17228 url=, key=/library/metadata/17228, containerKey=, metadataId=17228
Jul 01, 2017 16:39:03.031 [0x7fcad27fe700] DEBUG - Auth: We found auth token (xxxxxxxxxxxxxxxxxxxx), enabling token-based authentication.
Jul 01, 2017 16:39:03.031 [0x7fcad27fe700] DEBUG - Auth: authenticated user 728504 as tn
Jul 01, 2017 16:39:03.058 [0x7fcad2fff700] DEBUG - Auth: We found auth token (xxxxxxxxxxxxxxxxxxxx), enabling token-based authentication.
Jul 01, 2017 16:39:03.058 [0x7fcad2fff700] DEBUG - Auth: authenticated user 728504 as t*****n
Jul 01, 2017 16:39:03.489 [0x7fcaae3ff700] DEBUG - Play progress on 17228 ‘Serenity’ - got played 3555000 ms by account 728504!
Jul 01, 2017 16:39:03.628 [0x7fcaa2bf3700] DEBUG - Play progress on 17228 ‘Serenity’ - got played 3713000 ms by account 728504!
Jul 01, 2017 16:39:03.649 [0x7fcaaaffd700] DEBUG - Play progress on 17228 ‘Serenity’ - got played 3618000 ms by account 728504!
All looking good until the slow query and then an apparent loss of your local internet connection here.
Jul 01, 2017 16:39:31.007 [0x7fcaa9ffb700] DEBUG - NAT: UPnP, getPublicIP didn’t find usable IGD.
Jul 01, 2017 16:39:38.327 [0x7fcaa43f6700] WARN - SLOW QUERY: It took 780.000000 ms to retrieve 1 items.
Jul 01, 2017 16:39:41.010 [0x7fcaa9ffb700] WARN - NAT: PMP, timed out waiting for response.
Jul 01, 2017 16:39:41.010 [0x7fcaa9ffb700] DEBUG - MyPlex: Updating device connections (from timer: 0)
Jul 01, 2017 16:39:41.010 [0x7fcaa9ffb700] DEBUG - HTTP requesting PUT https://plex.tv/devices/3f78b6980c5e68f2ab2fbba083b5c4d1a03e6663?Connection[][uri]=http://10.55.55.12:32400&httpsEnabled=1&httpsRequired=0&X-Plex-Token=xxxxxxxxxxxxxxxxxxxx
Jul 01, 2017 16:39:41.726 [0x7fcaa9ffb700] DEBUG - HTTP 200 response from PUT https://plex.tv/devices/3f78b6980c5e68f2ab2fbba083b5c4d1a03e6663?Connection[][uri]=http://10.55.55.12:32400&httpsEnabled=1&httpsRequired=0&X-Plex-Token=xxxxxxxxxxxxxxxxxxxx
Jul 01, 2017 16:40:36.212 [0x7fcaa9ffb700] DEBUG - PublicAddressManager: Obtaining public address and mapping port.
Jul 01, 2017 16:40:36.212 [0x7fcaa3bf5700] DEBUG - PublicAddressManager: Obtaining public IP.
Jul 01, 2017 16:40:36.212 [0x7fcaa3bf5700] DEBUG - HTTP requesting GET http://plex.tv/pms/:/ip
Jul 01, 2017 16:40:36.232 [0x7fcaba3ff700] DEBUG - Sync: uploadStatus
One of you has a connectivity issue and from what I see here, your computer / modem is the root cause. I suggest start running some basic network stability and throughput tests ( Speedtest.net is a good basic starting point)