Add query time to logger

[timsehn]

This is a proposed fix for Dolt issue: dolthub/dolt#8909

I'm not certain if we want to just add this field or replace connect time. I think this implementation is safer.

[fulghum]

I like adding the new field, instead of replacing the connectTime field. People could probably also change their logger format to include the time, which is a good practice, and this would duplicate that, but could still be worth including.

Code changes looked fine, but Claude got pretty messy with the tests. 🤖