Heat 1/2 tablespoons oil in a large sauté pan over medium-high heat. Add onions, and sauté until well caramelized, 12-15 minutes. Remove from pan and let cool completely. Measure out 1/2 cup cooked onions and finely chop; set aside for sauce.
Combine cheese, remaining cooled onions,, and 1/2 tablespoons thyme; Season lightly and mix well.
Preheat ove to 375°F Lay chicken breasts, smooth side down on a cutting board. Cut a pocket in each breast with a sharp knife. Divide cheese and onion filling evenly between pockets. Cover with chicken fillet.
Heat remaining oil in same sauté pan over medium-high heat. Season chicken with salt and pepper and sear until golden brown. 1-2 minutes per side. Place in a roasting pan and cook 20 - 25 minutes or until the internal temperature is 170°F Let rest 5 minutes.
Return skillet to medium-high heat; pour maple syrup in pan to deglaze. Stir in broth, reserved caramelized onions and remaining thyme. Bring to a boil and reduce by half. Remove pan from heat, add any juices from the resting chicken and swirl in cold butter. Serve chicken breasts with maple sauce.