Lightly season the chicken tenderloins with salt and pepper, then dredge them in flour. This thin coating helps the chicken brown nicely and gives the sauce extra body later.
Heat olive oil and butter in a large skillet over medium heat. Cook the chicken until golden on both sides and fully cooked, then remove it from the pan and set aside.
In the same skillet, add sliced peppers and onions. Cook until softened and slightly caramelized, scraping up the flavorful bits left from the chicken.
Add minced garlic and red pepper flakes, stirring briefly until fragrant. Pour in the chicken broth and heavy cream, letting the sauce gently simmer and thicken.
Stir in freshly grated Parmesan until melted. Return the cooked chicken to the skillet and let it simmer in the sauce to soak up flavor.
While the sauce simmers, boil angel hair pasta until al dente. Drain well, reserving a little pasta water if needed to loosen the sauce.
Add pasta directly to the skillet and toss until evenly coated. Garnish with chopped parsley and extra Parmesan before serving.