Cheesy Baked Pasta with Cauliflower

Name: Cheesy Baked Pasta with Cauliflower
Ingredients: 1 pound pasta, such as medium shell or tube pasta; 1 (14-ounce) can cherry tomatoes, lightly crushed by hand; 8 ounces low-moisture whole-milk mozzarella, coarsely grated; 4 ounces provolone or other mildly sharp cheese (such as more of the cheddar below), coarsely grated; 2 ounces sharp cheddar, coarsely grated; 2 ounces Parmesan, grated; 1 1/2 cups heavy cream; Freshly ground black pepper; 1/2 head of cauliflower, cut into 1/2 pieces; Room-temperature butter or nonstick cooking oil spray (for pan)