Preparation

1. Preheat oven to 400°F and lightly grease a 9x13-inch baking dish with butter or nonstick spray. Set aside.
2. In a large skillet over medium-high heat, add the 2 tablespoons butter and 2 tablespoons olive oil. Once oil is shimmering, add onions and spread out evenly across the pan. Let cook, stirring occasionally, until onions are soft. If needed, reduce heat to medium to prevent burning.
3. Sprinkle onions with salt, sugar, and thyme, and continue cooking, stirring every few minutes, until onions are jammy and dark brown. If needed, add 1/2 cup beef broth to the pan to keep onions from drying out.
4. Once onions are caramelized, stir in balsamic vinegar and scrape up any stuck on bits from the bottom of the pan. Stir in garlic and cook until fragrant, 30 seconds to 1 minute. Remove from heat and set aside.
5. Cut a slit lengthwise into each chicken breast to form a pocket without cutting all the way through.
6. Stuff about 2 tablespoons of the caramelized onions into each pocket you created, followed by some of each of the cheeses. Seal shut with toothpicks.
7. Transfer remaining onions to the bottom of the prepared baking dish. Stir in remaining 1/2 cup beef broth. Set aside.
8. Heat remaining 2 tablespoons olive oil in a clean skillet over medium-high heat. Season chicken breasts with salt and pepper, then sear on each side until golden brown, about 4 minutes per side. (Chicken does not need to be cooked through at this point.)
9. Lay chicken on top of the caramelized onions in the baking dish. Spoon some of the onion sauce over each chicken breast, then top with any remaining cheeses. Bake until chicken is cooked through, 15-20 minutes more. Enjoy!