I have the same TimeoutError on Arch linux PC (openssl 1.1.1a).

It is random, but for now running test continuously at most 5 times successfully brings about the fail.

So I think msg335635 needs re-checking.
He saids PR 10011 makes this test pass in fedora 29 VM,
but thinking about that it only adds an conditional to FreeBSD, it is not likely.

    if sys.platform.startswith('freebsd'):
        client_context.options |= ssl.OP_NO_TLSv1_3

And indded, if commenting out the conditional line,
the test actually passes.
So I also think the commit needs re-considering.
f777fa5f2bd16ac8d60416eaa64eb9d2cf84ffac (Opt out of TLS 1.3 only on FreeBSD) 

---

$ uname -a
Linux **** 4.20.4-arch1-1-ARCH #1 SMP PREEMPT Wed Jan 23 00:12:22 UTC 2019 x86_64 GNU/Linux
(native, not venv nor vm)

$ ./python -m test.pythoninfo | grep ssl
ssl.HAS_SNI: True
ssl.OPENSSL_VERSION: OpenSSL 1.1.1a  20 Nov 2018
ssl.OPENSSL_VERSION_INFO: (1, 1, 1, 1, 15)
ssl.OP_ALL: 0x80000054
ssl.OP_NO_TLSv1_1: 0x10000000

$ ./python -X tracemalloc -m unittest -v test.test_asyncio.test_sslproto.SelectorStartTLSTests.test_start_tls_server_1
...
======================================================================
ERROR: test_start_tls_server_1 (test.test_asyncio.test_sslproto.SelectorStartTLSTests)
----------------------------------------------------------------------
Traceback (most recent call last):
  File "/home/philo/me/git/cpython/Lib/test/test_asyncio/test_sslproto.py", line 510, in test_start_tls_server_1
    self.loop.run_until_complete(run_main())
  File "/home/philo/me/git/cpython/Lib/asyncio/base_events.py", line 589, in run_until_complete
    return future.result()
  File "/home/philo/me/git/cpython/Lib/test/test_asyncio/test_sslproto.py", line 503, in run_main
    await asyncio.wait_for(
  File "/home/philo/me/git/cpython/Lib/asyncio/tasks.py", line 461, in wait_for
    raise exceptions.TimeoutError()
asyncio.exceptions.TimeoutError

----------------------------------------------------------------------
Ran 1 test in 60.076s