New AC blows much colder after starting/stopping engine

Hi,
I recently had the AC compressor/evaporator replaced in my vehicle by a reputable mechanic. The new AC is working, but not as it should be.

Here is the scenario:

1. I start the car on a warm day after the vehicle has been idle for many hours (overnight for example).
2. I begin driving and turn on the AC. The compressor comes on and the system starts blowing cool air. The car slowly begins to get cool, but not as cool as one would expect with a new compressor.
3. I drive for 10-15 minutes and the temperature of the air coming out of the vents has stabilized and is cool, not getting any colder.
4. Next I stop the vehicle and shut off the engine for about 5 minutes (get gas, run into the store for an errand).
5. When I restart the vehicle after about 5 minutes the AC immediately begins blowing really cold air and the car quickly cools to a point where I have to back off the temperature of the AC or shut if off altogether.

My question is why do I not get very cold from the AC when I start the car and only after I stop/restart it for a few minutes?

I have been able to repeat this several times with the vehicle so I know it is not a one-off situation.

Thanks,
RKU