Utility sink continues to dribble from hot faucet after closing hot water line
Depending on the size and length of the piping throughout the house it's not unreasonable to estimate that there might be two gallons or more water contained in just the hot water piping. In the case of copper tubing, 50 feet of 3/4&quot; contains 1.1 gal while 80 feet of 1/2&quot; contains another 0.9 gal. It's relatively easy to consume that much piping especially in a two-story house or multi-bath house.
After the hot piping is drained down (whatever its volume may be), if water continues to exit from that basement-level faucet, it is being replenished from somewhere. Some valve is leaking water from the cold system into the hot system. Two possibilities come to mind:

The shutoff at the water heater leaks, allowing new water to flow through the heater tank to the faucet.
A mixing valve somewhere in the house is allowing water to leak from the cold side to the hot side, but not leaking to the faucet's normal output.

You mentioned testing whether this faucet's mixer leaks by turning off its cold supply. Try turning off all the cold supplies at all the other single-handle faucets in the house too. If the drip stops, then turn those supplies back on one at a time to find one that leaks. There could even be more than one that leaks. In the event that you have one or more single-handle faucets for a bathtub filler or shower (where there's no secondary shutoff valve as there would be under a sink) you might not be able to prove whether or not one of them has such a cross-supply leak.