Cook chicken, stirring occasionally, until the outer part is no longer pink. Remove chicken with a slotted spoon. Lower the heat to medium. Add onion and red pepper to the skillet, cooking until the onion is translucent and the peppers are tender. Include garlic and cook briefly. Return chicken to the pan with onions, peppers, and garlic. Cover and cook until chicken is fully cooked.
For the sauce, heat heavy cream and butter in a saucepan over medium heat until the butter melts. Mix in parmesan cheese until combined. Stir often to prevent scorching. Once it gently boils, reduce heat and let it thicken. Season with salt and pepper. Serve by adding chicken and vegetables to the pasta and pouring the sauce over.