Cook the Pasta: Get a large pot of salted water to a rolling boil. Add the rigatoni and cook according to package directions, but pull it out when it's just shy of al dente—it will finish cooking in the sauce. Before you drain it, reserve about a cup of the starchy pasta water. Drain the pasta and set it aside.
Sear the Chicken: While the pasta cooks, pat your chicken pieces completely dry with a paper towel and season them generously with salt and pepper. Heat the olive oil and one tablespoon of butter in your Dutch oven over medium-high heat. Once the butter is foaming, add the chicken in a single layer. Don't crowd the pan! Work in batches if you need to. Sear the chicken for 2-3 minutes per side, just until it's beautifully golden brown. It doesn't need to be cooked through. Remove the browned chicken to a plate and set it aside.
Sauté the Aromatics: Reduce the heat to medium and add the remaining tablespoon of butter to the pot. Add the chopped onion, bell peppers, and mushrooms. Cook, stirring occasionally, for about 5-7 minutes, until the onions are soft and translucent and the mushrooms have released their water. Now for the flavor bomb: stir in the minced garlic and the chopped hot cherry peppers. Sizzle for another minute, just until that nutty garlic aroma starts to perfume your entire kitchen.
Build the Sauce: Pour in the chicken broth to deglaze the pot. Use your wooden spoon to scrape up all those delicious browned bits from the bottom—that's pure flavor! Stir in the crushed tomatoes and season with a pinch of salt and pepper.
Simmer and Thicken: This step is critical. Bring the sauce to a gentle simmer and let it cook, uncovered, for 10-15 minutes. You'll watch as the sauce visibly thickens and deepens in color. This is how you avoid a watery mess and build a concentrated, robust tomato base. Your patience will be rewarded.
Make It Creamy: Reduce the heat to low and slowly pour in the heavy cream, stirring constantly until it’s fully incorporated. The color will transform into that iconic, vibrant pinkish-orange. Stir in the grated Pecorino Romano cheese until it melts smoothly into the sauce.
Combine and Finish: Now it's time to bring it all together. Return the seared chicken and any juices from the plate back into the pot. Add the drained rigatoni, the black olives, and two tablespoons of the hot cherry pepper brine. Stir everything gently until the pasta and chicken are coated in that gorgeous sauce. Let it all simmer together for another 2-3 minutes, allowing the pasta to absorb the flavor and finish cooking. If the sauce seems too thick, add a splash of that reserved pasta water until it reaches the perfect consistency.
Serve Immediately: Ladle the Chicken Riggies into warm bowls. Garnish with a little extra Pecorino cheese and some fresh parsley, if you like. Get it to the table while it's hot.