Heat 1 tablespoon oil and the butter over medium heat in a large oven-proof skillet.Add onion and cook until golden, about 8 to 10 minutes, stirring frequently so they don't burn.
Add garlic, thyme, and red pepper flakes and cook 1 minute. Add beef broth and cook another minute. Season to taste with salt and pepper.Transfer onions to a medium bowl and set side.
Cut a pocket in each chicken breast. Season chicken with salt and pepper.
Add mozzarella and Parmesan cheese to bowl with onions and stir to combine.
Stuff the pockets in the chicken with onion/cheese mixture. Secure with toothpicks if necessary.
Heat remaining oil in the same skillet over medium-high heat. Add chicken and cook for 3 minutes per side.
Transfer pan to oven and cook for 8 to 10 minutes or until cooked through.