Heat the oil and butter in a large (12-inch), 2-inch deep skillet over medium heat. When the butter stops foaming, stir in the onion and garlic until softened and fragrant, 2-3 minutes.
Stir in the chicken and cook for 1 minute, then sprinkle salt and flour over the chicken, stirring to coat.
Gradually stir in the broth, and then the cream. Raise the heat and bring to a boil.
Stir in the pasta noodles. Bring the heat down to a low simmer and cover the pan. Cook 12-15 minutes until pasta is tender.
Remove from heat and stir in the pesto until incorporated. Remove from heat, top with parmesan cheese and serve immediately.
