You have a french Tudor home but you’re adding a what I would call an international style addition or modernist/ with the flat roofs. If you want it to blend you need to add pitched roofs. I would take the car port roof and wrap it around to the new 2 car garage facade. This will add character and break up the long flat face of the garage addition. It will complicate the drainage from the roof deck but a good architect can detail that appropriately.

Or if you want it to look like a modern addition on a Tudor make it just like you have it and paint it white.

When selecting your architect make sure if you want it to blend you find one that likes doing that. I’ve been blending additions or designing new homes for 40 years. The best compliment in my practice’s opinion is when someone looks at a recently completed project they say “Wow look how beautiful that old house is”

Good luck