Why does it take a second for water to come out of the kitchen faucet?
Over the weekend I determined what the cause was. Thanks to everyone for the helpful suggestions; the actual problem was slightly different.

The problem was that the sprayer hose was loose on the sprayer end. Since the hose is pressurized when the faucet is on, this meant that the sprayer leaked a little from the bottom while the water was on, and a little bit after turning the faucet off. This meant that the next time you turned the faucet on, the hose was partially empty of water, so it took a second to pressurize the hose before water would come out of the faucet.

Tightening the sprayer hose connection seems to have fixed the problem.