[3.11] gh-109709: Fix asyncio test_stdin_broken_pipe() (#109710) (#109735)

gh-109709: Fix asyncio test_stdin_broken_pipe() (#109710)

Replace harcoded sleep of 500 ms with synchronization using a pipe.

Fix also Process._feed_stdin(): catch also BrokenPipeError on
stdin.write(input), not only on stdin.drain().

(cherry picked from commit cbbdf2c1440c804adcfc32ea0470865b3b3b8eb2)