Cook Pasta: Bring a large pot of salted water to a boil. Cook the pasta according to package directions until al dente. Drain, reserving ½ cup of pasta water, and set aside.
Prepare Peppers: Peel the red bell peppers with a vegetable peeler (or roast and peel). Remove the seeds and cut into large pieces.
Sauté Aromatics and Chorizo: Heat the olive oil in a large, deep skillet over medium heat. Add the chopped onion and garlic and sauté for 2-3 minutes until softened. Add the sliced chorizo and cook for about 3 minutes, until it starts to brown and release its oils.
Cook Peppers: Add the prepared red pepper pieces to the skillet. Simmer over medium heat for 10-15 minutes, stirring occasionally, until the peppers are tender. Season with the paprika, cayenne pepper, salt, and pepper.
Make Sun Dried Tomato Paste: While the peppers cook, place the sun dried tomatoes (with their oil), pine nuts, and Parmesan cheese in a food processor. Season with a pinch of salt and pepper. Blend until a thick paste forms.
Combine Sauce: Stir the sun dried tomato paste into the skillet with the chorizo and peppers. Cook for 1-2 minutes. Add the halved cherry tomatoes and mozzarella balls. Simmer for 5 minutes, until the tomatoes soften slightly and the mozzarella is warm.
Toss with Pasta: Add the cooked pasta to the skillet. Toss everything together until the pasta is well coated. If the sauce is too thick, add a splash of reserved pasta water to loosen it.
Serve: Serve immediately, garnished generously with chopped fresh basil.