Bring a large pot of well-salted water to a boil. Cook the rigatoni until just shy of al dente according to the package timing, reserve 1 cup of pasta water, and drain.

Season the chicken pieces with about 1/2 teaspoon of the kosher salt and the black pepper. Heat 1 tablespoon olive oil in a large skillet or Dutch oven over medium-high heat, then cook the chicken in a single layer for about 5–7 minutes, until browned and cooked through; transfer it to a plate.

Add the remaining tablespoon of olive oil to the pan. Sauté the onion and red bell pepper for 4–5 minutes, until softened.

Add the garlic, chopped hot cherry peppers, 1–2 tablespoons of pepper brine, crushed red pepper flakes, and Italian seasoning. Cook for about 30 seconds, just until fragrant.

Pour in the crushed tomatoes and chicken stock. Bring the mixture to a gentle simmer and cook for 10–15 minutes, stirring occasionally, until the sauce is slightly thickened.

Reduce the heat to low and stir in the heavy cream. Keep the sauce from reaching a vigorous boil after the cream is added.

Return the chicken to the pan, then add the drained rigatoni and toss until coated. Stir in the Parmesan, adding reserved pasta water a few tablespoons at a time if the sauce needs loosening.

Taste and adjust the salt and heat. Finish with chopped parsley or basil and serve with additional Parmesan.