Prepare the Chicken: Heat 2 tablespoons of olive oil in a large skillet over medium-high heat.
Add the chicken pieces, season lightly with salt and pepper and cook until browned on both sides and cooked through. Remove from the skillet, set aside to cool for at least 10 or 15 minutes and then chop up into small bite sized pieces.
Saute the Vegetables: In the same skillet, add the remaining 2 tablespoons of olive oil.
Add the chopped onion and bell peppers. Sauté until they start to soften and the onions become a bit translucent. (optional can add mushrooms and olives here if you are using them)
Add Flavor and Tomatoes: Stir in the minced garlic and cook for about 1 minute or only until fragrant. Do not brown at all.
Add the tomato paste and cook, stirring, for another 2 minutes to cook out the raw flavor of the paste.
Pour in the wine, whisk or combine well, and let it simmer for a couple of minutes, allowing the alcohol to evaporate.
Add Tomatoes and Chicken: Add the crushed tomato sauce, chopped hot/cherry peppers, and chopped chicken to the skillet. Stir to combine.
Season with pepper to taste. Remember that the cherry peppers can add some heat, so adjust according to your preference.
Simmer and Combine: Reduce the heat to medium-low and let the sauce simmer for at least 15-20 minutes with lid ajar or longer until it's reached your desired thickness. This allows the flavors to meld and the sauce to thicken slightly. Don't forget to stir intermittently.
Add Cheese and adjust taste: Once the sauce has thickened, stir in the grated Parmesan cheese and allow it to melt into the sauce, creating a creamy texture. OPTIONAL: Add heavy cream (suggested 1/4 cup or add enough until you get. a pink/blush sauce).
Final Touches: If the sauce seems too thick, you can add a bit of reserved pasta water to reach your desired consistency.
Taste and adjust seasoning if needed. You can also adjust the spiciness by adding more cherry peppers if desired and you can add a bit more brine from the hot peppers, slowly, if desired (1 tablespoon at a time).
Serve: Serve the Chicken Riggie sauce over cooked pasta and sprinkle with additional cheese if desired.