Preheat oven to 400°F.
Make the onions. Heat 2 tbsp olive oil in a pan over medium heat. Add the onions and cook for 20-25 minutes until soft, golden, and caramelized, stir them often throughout the process to ensure even cooking. Remove from the pan and set aside
If using whole chicken breasts, slice them in half. Tenderize with a meat mallet and season with salt. Add the remaining 1 tbsp olive oil to the pan. Sear the chicken over medium-high heat until golden on both sides (about 5 -6 minutes each side) Remove from the pan.
Return the onions to the pan and add the garlic. Cook for 1 minute until fragrant. Pour in the beef broth, then add the balsamic vinegar, Worcestershire sauce, thyme, salt, and pepper. Bring to a simmer.
In a small bowl, mix the cornstarch with 1 tbsp of liquid from the pan to form a slurry. Reduce the heat to low and stir the slurry into the sauce and cook for a couple of minutes until it thickens.
Return the chicken to the pan, spoon the sauce over the top, and sprinkle with Gruyère.
Bake for 10 minutes, or until the cheese is golden and crispy. For extra crispiness, broil for 1-2 minutes at the end.