Heat 2 tablespoons of olive oil in a pan over medium heat. Add half of the chopped onion and garlic, cooking until translucent.
Add the lobster shells to the pan, and cook on high heat. Crush the shells as they cook to release maximum flavor.
Deglaze the pan with white wine, letting the alcohol cook off and the wine reduce for 2–3 minutes. Add 1/2 cup of water, Old Bay seasoning, paprika, salt, and black pepper. Let this simmer for 3 more minutes.
Strain the mixture to remove and discard the shells, reserving the flavorful lobster sauce.
In the same pan, melt the butter and add the remaining onion and garlic. Cook for 1–2 minutes until softened.
Pour in the cooking cream and add the mozzarella. Cook, stirring, for another 2 minutes until smooth.
Add the lobster sauce to this creamy mixture, stirring to combine. Cook for 1 more minute, then add the parmesan cheese. Stir well, then remove the pan from heat and set the sauce aside