Uses 2 channels to synchronize that test, because
relying on sleeps creates flaky behavior, thus:

a) 1 buffered channel to send back the last spurious line
without having to reason about "happens before" behavior
a) 1 buffered channel at the end of the handler; it'll
be controlled by whether we expect to timeout or not,
but will always be closed when the test ends

Fixes #35051

Change-Id: Iff735aa8d1ed9de8d92b792374ec161cc0a72798
Reviewed-on: https://go-review.googlesource.com/c/go/+/208477
Run-TryBot: Emmanuel Odeke <emm.odeke@gmail.com>
TryBot-Result: Gobot Gobot <gobot@golang.org>
Reviewed-by: Bryan C. Mills <bcmills@google.com>