Bring a large pot of salted water to a boil. Cook the pasta according to package directions. Drain and set aside.
Season the chicken pieces with salt and pepper. In a bowl, dredge the chicken in flour and shake off the excess. Heat the butter and the olive oil in a large pan over high medium heat. Once the butter melts, add the chicken pieces and the garlic. Cook for 6 minutes or so, until the chicken is cooked through and golden. Transfer to a plate and set aside.
Add the heavy cream to the same pan and bring to a simmer. Add the grated cheddar cheese and cook until most of it has melted. Add the pasta and toss it around so everything gets coated with the sauce. Add the chicken and carefully toss to combine everything. Turn off heat and garnish with parsley.