Boil a large pot of lightly salted water, then add pasta and cook according to insturctions on box for al dente.
Meanwhile, slice the chicken breasts into strips, and season with salt and pepper.
Then heat the butter and olive oil in a large skillet oven medium heat until the butter is completely melted.
Add in the fresh minced garlic and saute for about 1 minute before adding chicken strips.
Cook until chicken is completely white on the outside.
Reduce heat, then stir in the heavy cream, pesto, Parmesan cheese, and a dash of salt and pepper.
Cook until chicken is no longer pink on the inside.
Drain pasta, and add it to the chicken and sauce.
Mix well and serve, sprinkling some more Parmesan cheese on top.