In a large oven-safe skillet, heat the olive oil over medium heat. Season the chicken breasts with salt and pepper, then add them to the skillet. Sear the chicken for about 4-5 minutes on each side until golden brown. Remove the chicken from the skillet and set aside.
In the same skillet, add the sliced onions and cook for about 10-12 minutes, stirring occasionally, until they are caramelized and golden brown. Add the minced garlic and thyme, cooking for an additional minute until fragrant.
Pour in the beef broth and balsamic vinegar, scraping up any browned bits from the bottom of the skillet. Bring the mixture to a simmer.
Return the chicken to the skillet, spooning some of the onion mixture over the top. Sprinkle the shredded Gruyère cheese evenly over the chicken.
Transfer the skillet to the preheated oven and bake for 20-25 minutes, or until the chicken is cooked through (internal temperature of 165°F) and the cheese is bubbly and golden.
Remove from the oven and let it rest for a few minutes. Garnish with chopped parsley before serving.
Notes
For a richer flavor, use a combination of beef and chicken broth.
Add a splash of white wine to the onion mixture for an extra depth of flavor.