You sure flipping frequently makes the internal temp rise slower? Kenji found flipping burgers frequently reduced the time to reach his desired temp by a third. https://aht.seriouseats.com/2010/02/the-burger-lab-how-many-times-should-you-flip-a-burger-while-cooking.html

Mozzarella is nice, but it’s really easy to buy good mozzarella. How about making halloumi, which is delicious and much harder to get?