Bring a large pot of heavily salted water to a boil. Season the chicken pieces with the salt and black pepper.
Heat the olive oil and butter in a large skillet or Dutch oven over medium-high heat. Brown the chicken in a single layer, working in batches, about 4 to 5 minutes. It does not need to cook through. Transfer to a plate.
Reduce the heat to medium. Add the onion, bell pepper, and mushrooms to the same pan. Cook 7 to 8 minutes, until softened and the mushroom liquid cooks off. Stir in the garlic and tomato paste and cook 1 minute more.
Pour in the sherry and scrape up the browned bits from the pan. Reduce by half, about 2 minutes. Add the rigatoni to the boiling water now and cook 1 minute shy of al dente. (NOTE: If the pasta finishes before the sauce is ready, drain it and toss with a little olive oil so it doesn't stick.)
Stir in the crushed tomatoes, cherry peppers, cherry pepper brine, Italian seasoning, and hot paprika. Simmer 8 to 10 minutes. You can simmer longer if you'd like, 20-30 minutes (adjust pasta cooking time as needed).
Lower the heat so the sauce barely bubbles. Stir in the heavy cream, then return the chicken and any accumulated juices. Simmer 5 minutes, until the chicken cooks through and the sauce thickens enough to coat a spoon.
Remove the pan from the heat and stir in the Pecorino a handful at a time so it melts smoothly instead of clumping. Return the pan to low heat, add the olives and the drained rigatoni, and toss for 1 to 2 minutes, loosening with reserved pasta water a splash at a time, until the pasta is al dente.
Taste and adjust salt and pepper brine. Top with torn basil and serve with extra Pecorino and red pepper flakes.