Preheat oven to 400°F (200°C). Lightly oil a 2- to 3-quart baking dish.
Brown the sausage: Heat a large skillet over medium-high heat. Add sausage, breaking it apart with a spoon. Cook until browned, 5–7 minutes. Drain excess fat if needed, leaving about 1 tablespoon in the pan.
Add vegetables: Stir in onion and bell peppers. Sauté until the onion turns translucent and the peppers begin to soften, about 5–6 minutes.
Add garlic and spices: Stir in minced garlic, paprika, oregano, basil, and red pepper flakes. Cook 30 seconds until fragrant.
Build the sauce: Pour in crushed tomatoes and broth. Stir and scrape the bottom of the pan to release any browned bits. Simmer 3–5 minutes until slightly thickened. Season with salt and pepper.
Make it creamy: Remove from heat. Stir in ricotta and ½ cup mozzarella until just combined.
Assemble: Spread mixture evenly into the prepared baking dish. Top with remaining mozzarella, provolone, and Parmesan.
Bake: Bake uncovered for 15–20 minutes, or until the cheese is melted and bubbling around the edges.
Finish and serve: Drizzle vinegar on top, let rest 5–10 minutes, then sprinkle with chopped parsley before serving.