In a large skillet, heat olive oil and butter over medium heat. Add the sliced onions and cook, stirring frequently, for about 15-20 minutes, until they are caramelized and golden brown. Add the minced garlic and cook for an additional 1-2 minutes until fragrant.
Season the chicken breasts with salt, pepper, and thyme. Push the caramelized onions to the side of the skillet and nestle the chicken breasts in the center. Cook for about 5 minutes on each side until the chicken is browned.
Pour in the beef broth and balsamic vinegar, scraping the bottom of the skillet to deglaze and incorporate the flavors. Bring to a gentle simmer, then cover and reduce the heat to low. Let it cook for about 15-20 minutes, or until the chicken is cooked through (internal temperature of 165°F).
Once the chicken is done cooking, sprinkle the shredded Gruyère cheese over the chicken and onions. Cover the skillet and let the cheese melt for about 2-3 minutes.
Carefully plate the chicken along with the caramelized onions and melted cheese. Garnish with fresh parsley before serving.