Heat oil in a large heavy saucepan over medium heat. Add flour and cook, whisking constantly, for 1 to 2 minutes. Add hot milk and bring to a simmer, whisking, until smooth and slightly thickened, 3 to 4 minutes. Add nutmeg and season with salt and pepper. Transfer to a large bowl; let cool for 5 minutes. Add eggs, mozzarella, ricotta, prosciutto (or ham) and 2 tablespoons Parmesan. Mix well.