Cook the rigatoni in a large pot of salted boiling water according to package directions until al dente. Drain and set aside.
While the pasta cooks, melt the butter in a large pot or Dutch oven over medium-high heat. Add the chicken pieces in a single layer and cook for 3-4 minutes, until lightly browned. It does not need to be cooked through at this stage.
Add the sliced mushrooms, chopped onion, and green bell pepper to the pot. Cook, stirring occasionally, for 5-7 minutes, until the vegetables have softened.
Stir in the chopped hot cherry peppers, paprika, a pinch of salt, and a few grinds of black pepper.
Pour in the chicken broth, tomato sauce, and heavy cream. Stir everything together and bring the mixture to a simmer. Reduce the heat to low, cover, and let it gently simmer for 15 minutes, stirring occasionally. The chicken should be cooked through to an internal temperature of 165°F (74°C).
Stir in the sliced black olives.
In a small bowl, whisk together the cornstarch and cold water until smooth. Pour the slurry into the simmering sauce while stirring constantly. Continue to cook and stir for 1-2 minutes until the sauce has thickened.
Add the cooked rigatoni to the pot with the sauce. Stir gently to coat all the pasta. Serve immediately, topped with grated Parmesan cheese.