In a medium skillet, melt the butter with 2 tablespoons of the olive oil. Add the sausage and cook and crumble over medium heat until sausage loses its pinkness, 5 to 7 minutes. Add the garlic and cook for 30 seconds more. Add the mushrooms and cook, stirring often, until softened, about 2 minutes. Add the onion and cook until softened, about 5 minutes.
Transfer this mixture to a large bowl and add the pasta, that is already mixed with the tomato sauce (or unsauced), prosciutto, cream, and a 1/4 cup of the Parmigiano Reggiano cheese, and salt and pepper, to taste. Add the eggs and stir to combine.
Heat the remaining olive oil in a clean 10-inch, non-stick ovenproof skillet over medium heat (if the skillet is not ovenproof, wrap the handle tightly in aluminum foil). Add the egg mixture and cook over medium heat until the bottom of eggs is set and a light golden brown color appears. (Do NOT over-brown by using too much heat, too fast!).
Place the pan into the preheated oven and bake for 7 minutes, or until eggs are set to desired consistency and begin to puff up a bit. Remove from oven and let rest for about 2 minutes. Cut into wedges and serve with remaining grated cheese, and warm red sauce and sprinkled with shredded basil leaves, if desired.