Season the chicken on both sides with salt and pepper.
Heat the oil in a saucepan over medium heat. Add garlic, peppers, onion; sauté, stirring frequently until soft for about 3-5 minutes. Then add the flour and Cajun seasoning. Cook until the flour turns golden brown.
Gradually stir in the broth, thyme, and lemon zest. Cover with a lid and bring to a low simmer for 15 minutes or more to build flavor. Taste and adjust the seasoning as needed. Add in the cream and bring to a simmer for 3 minutes. Remove the thyme.
In a large pot of boiling salted water, cook pasta according to package instructions; drain well.
In the meantime, heat the oil in a sauté pan over medium heat. Add chicken and cook on both sides until golden brown to an internal temperature of 160°F using a thermometer. Then set aside to rest for 2-3 minutes before slicing.
Toss the pasta in the sauce and serve immediately with chicken so the pasta doesn't soak up all the sauce. Garnish with scallions. Serve with dinner rolls.
Notes
Pro tips: Butterfly the chicken: Lay the breasts down on a cutting board, using a sharp knife parallel to the cutting board slice the breasts through the middle (carefully) all the way in half.