Preheat the oven to 400°F (200°C).
In a large skillet over medium heat, melt the butter and 1 tablespoon of olive oil. Add the sliced onions and cook, stirring occasionally, for 25 to 30 minutes until deep golden brown.
Stir in the balsamic vinegar and beef broth. Simmer for 2 to 3 minutes until the liquid reduces by half, then remove from heat.
Season the chicken breasts on both sides with salt, pepper, and dried thyme.
In a separate oven-safe skillet or pan over medium-high heat, add the remaining tablespoon of olive oil. Sear the chicken for 3 minutes per side until golden.
Spread the caramelized onions evenly over the top of each chicken breast.
Sprinkle the shredded Gruyère and Provolone cheeses over the onions.
Transfer the skillet to the oven and bake for 15 to 20 minutes, or until the chicken reaches an internal temperature of 165°F (74°C).
Optional: Place under the broiler for 1 to 2 minutes until the cheese is browned and bubbly.