In a large skillet, heat the olive oil over medium heat. Add the sliced onions and sprinkle with salt. Cook the onions slowly, stirring occasionally, until they are caramelized and golden brown, about 15-20 minutes.
While the onions are cooking, season the chicken breasts with black pepper and dried thyme. In a separate pan, sear the chicken over medium-high heat for 5-6 minutes on each side until browned.
Once the onions are caramelized, add the balsamic vinegar and beef broth to the pan with the onions. Stir to combine and bring to a simmer.
Nestle the seared chicken breasts in the onion mixture, cover, and let simmer for 10-15 minutes until the chicken is cooked through (internal temperature should reach 165°F).
Preheat the broiler in your oven. Once the chicken is cooked, sprinkle the shredded Gruyère cheese evenly over the chicken and onion mixture.
Place the skillet under the broiler for 2-3 minutes, or until the cheese is melted and bubbly.
Remove from the oven, garnish with fresh parsley if desired, and serve hot.