Bring a large pot of lightly salted water to a boil over high heat. Add rigatoni pasta and cook until tender yet firm to the bite, 8 to 10 minutes. Drain and keep warm.
Heat olive oil in a large skillet over medium-high heat. Season chicken with salt and pepper, and cook in the hot oil until lightly browned on all sides, and no longer pink in the center, about 7 minutes. Remove the chicken from the skillet and keep warm.
Stir onion, Cubanelle peppers, and garlic into the skillet. Cook and stir until onion softens, about 4 minutes. Add crushed tomatoes, roasted red peppers, and hot cherry peppers. Bring to a simmer. Stir in cooked chicken and heavy cream; simmer for 2 to 3 minutes. Stir in cooked pasta.