This builds on 753d61b. Before we send a request, we can do a 1-byte
nonblocking peek on the connection. If the server has closed the
connection, this will give us an EOF, and we can take the connection out
of the pool before sending any request on it. This will reduce the
likelihood that we send a non-retryable POST on an already-closed
connection.
The server could still potentially close the connection between when we
make this check and when we finish sending the request, but this should
handle the majority of cases.
