Definitely have the dealer check out the car. It sounds almost like a loose or bad motor mount. Most of the time I don't notice the ICE starting/stopping. I would say starting from a stop is when it is most noticeable. Even then it is mild, like an automatic transmission shifting from 1st to 2nd. Just wondering, how is the engine oil level?

It might be worth opening the hood and watching/listening to the ICE startup. Just open the hood, press the Start button, and about 7 seconds later the ICE will start. You can "force" the ICE to start at will by putting the transmission in Park, and pressing the gas pedal. Release the pedal and the ICE will stop after a short time. Repeat as needed to observe how the ICE starts and listen for unusual noises, etc.

If you know anyone else with a TCH, compare how theirs starts, take them for a drive and get their opinion.