In a large pot of salted boiling water, add the cavatappi pasta. Cook until al dente following package instructions (approximately 8-10 minutes). Reserve 1 cup of pasta water, then drain the remaining and set aside.
Season the chicken evenly with smoked paprika, onion powder, garlic, and crushed red pepper.
In a large skillet over medium-high heat, add olive oil and butter. Once the butter is melted, add the chicken. Cook for 5-7 minutes per side or until fully cooked, reaching an internal temperature of 165°F. Transfer the chicken to a plate and tent to keep warm.
For the sauce, in the same skillet, add butter. Once melted, add onions and sun-dried tomatoes. Cook until softened (about 3-5 minutes). Add garlic and cook for an additional minute.
Add chicken broth, stirring with a spatula to break up any browned bits in the pan.
Incorporate heavy cream, cream cheese, parmesan, cheddar, and Monterey Jack cheese. Stir until all cheeses are melted and combined. Adjust the sauce's thickness by gradually adding reserved pasta water until the desired consistency is reached.
Add the cooked pasta to the sauce. Season to taste with salt and pepper, stirring to combine.
Slice the chicken and place it on top of the pasta.
Serve warm, garnished with chopped parsley.