Cook rigatoni in a large pot of boiling salted water according to package directions until al dente (usually about 5 to 7 minutes). Reserve 1 cup of pasta water, then drain the pasta. Set aside pasta and reserved pasta water.
Season chicken with salt and pepper. Heat olive oil over medium-high heat. Cook the chicken, stirring occasionally until lightly browned on all sides, about 5 to 6 minutes. It's okay if it's not cook all the way through yet. Transfer to a plate and set aside.
Add onion and cook until translucent, about 5 minutes. Add bell peppers and hot or sweet pickled peppers. Saute for another 5 to 6 minutes until the peppers have softened. Add garlic during the last minute.
Deglaze the pan with ½ cup white wine or water for 2-3 minutes, scraping up any browned bits off the bottom of the pan. That's where the flavor is!
Stir in crushed tomatoes, Italian seasoning, and red pepper flakes, if using. Bring to a simmer. Reduce heat to medium and cook for 5 to 7 minutes while stirring to allow the flavors to blend.
Return the chicken and any accumulated juices to the pan. Cook for another 3 to 4 minutes, stirring occasionally, until the chicken is cooked through. Stir in the parmesan cheese.
Add the cooked pasta and ½ cup of the reserved pasta water, stirring until coated. Add a splash or two of additional pasta water as needed to get a consistency you like. Simmer 1 to 2 minutes until done. Garnish with more cheese, fresh basil or parsley, and additional red pepper flakes, if desired.