Heat olive oil in a large skillet over medium heat. Immediately add shrimp and garlic; cook until shrimp is pink, 3 to 4 minutes. Sprinkle salt, basil, and paprika over shrimp. Cook and stir for 1 minute more, leaving shrimp slightly undercooked to finish cooking in the sauce.

Remove shrimp mixture from the skillet. Add mushrooms to the same skillet. Cook and stir, seasoning with salt, until juicy, about 5 minutes. Add the shrimp; immediately pour in half-and-half, mozzarella cheese, and Parmesan cheese. Bring to a boil. Reduce heat and simmer, constantly stirring, until cheese melts, 5 to 8 minutes. Cover with a lid and remove from heat.