Melt the butter and sauté onions: In a large skillet, melt 4 tablespoons of butter over medium heat. Add 2 pounds of sliced yellow onions and cook for 45 minutes to 1 hour until golden brown.

Add garlic and thyme: Stir in 2 cloves of chopped garlic and 1 teaspoon of chopped fresh thyme, cooking for an additional minute. Sprinkle 2 tablespoons of flour over the mixture and stir to combine.

Incorporate the broth: Pour in half of the 2 cups of broth, bringing to a boil. Reduce heat and let simmer for 5 minutes until sauce thickens slightly.

Brown the chicken: In a separate pan, heat 1 tablespoon of olive oil over medium-high heat. Season 1 pound of chicken with salt and pepper and brown for 2 to 4 minutes on each side.

Create the sauce: Deglaze the chicken pan with remaining broth, stir in 1 tablespoon of balsamic vinegar and 2 tablespoons of Dijon mustard, and bring to a boil. Let reduce for 5 minutes.

Assemble the dish: Layer caramelized onions in an oven-safe dish, place browned chicken on top, pour sauce over everything, cover with aluminum foil, and bake at 350°F (180°C) for 20 minutes.

Broil for the cheese: Uncover and sprinkle 1 cup of grated Gruyere cheese over the chicken. Broil for 2 to 4 minutes until cheese is melted and slightly golden.