Heat 1/2 tbsp (7 mL) oil in a large sauté pan over medium-high heat. Add onions and sauté until well caramelized, 12 to 15 min. Remove from pan and let cool completely. Measure out 1/2 cup (125 mL) cooked onions and finely chop; set aside for sauce.
Combine cheese, remaining cooled onions and 1/2 tbsp (7 mL) thyme; season lightly and mix well.
Preheat oven to 375°F (190°C). 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 filet.
Heat remaining oil in same sauté pan over medium-high heat. Season chicken with salt and pepper and sear until golden brown, 1 to 2 min. per side. Place in a roasting pan and cook 20 to 25 min., or until the internal temperature is 170°F (77°C). Let rest 5 min.
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.