Bring a pot of salted water to a boil (we usually add about 1/4 cup of salt). Add the pasta and cook until very al dente, 10 minutes. Drain the pasta, toss it with olive oil and set aside.
Preheat the oven to 450°. In a large mixing bowl, combine the four cheeses. Remove one half of the cheese and set aside.
Pour the tomatoes, mushrooms, and milk into the mixing bowl with half the cheese. Season the mixture with salt and pepper and stir to combine. Add the cooked pasta and rosemary and toss.
Pour the pasta into a shallow 12-inch skillet. Sprinkle with the remaining ½ of the cheese and bake until the top is bubbling and golden brown, 10-15 minutes.