Change the connected time from time_t to timespec.

The primary gain as the code stands is that we become immune to
issues with the clock going backwards etc. in logs (since we can
now use a monotonic timer).

However, the motivating change is that we will soon be implementing request
read timeouts. At that point, not only will the clock data be much more
important to get right, but it will also be nice to have more fine-grained
timestamps to be able to locate clients semi-uniquely in a sorted list.