In a soup pot or Dutch oven, add 1 tablespoon oil and sausage and cook over medium heat until lightly browned, breaking it into small chunks with a wooden spoon, 5-6 minutes. Remove sausage from pot and set aside.
Add remaining 2 tablespoons olive oil to pot and sauté the onion until golden, about 5 minutes.
Add the peppers, season with salt, and continue cooking until softened, 7 to 8 minutes.
While the peppers are cooking, fill a pot for the pasta with water, place over high heat and bring to a boil.
To the sauce, add tomato paste and water (or tomato sauce if using), chopped rosemary and oregano. Stir in cooked sausage and turn heat down to low.
When the sauce is ready and the water is boiling, cook pasta until al dente. Reserve 1 cup pasta water then drain well and toss with the sauce along with the butter and grated Parmigiano-Reggiano. Stir in ½ cup pasta water. Add a little more pasta water if needed just until the sauce is glossy, then plate and serve topped with more grated Parmigiano-Reggiano.