Preheat your oven to 375°F (190°C).
Heat olive oil and butter in a skillet over medium heat.
Add sliced onions and cook until caramelized, about 15 minutes.
Stir in thyme, salt, pepper, and balsamic vinegar.
Remove onions from skillet and set aside.
Season chicken breasts with salt and pepper.
In the same skillet, sear the chicken for 4-5 minutes on each side until golden brown.
Pour beef broth over the chicken and let simmer for 5 minutes.
Top each chicken breast with caramelized onions and Gruyère cheese.
Bake in the preheated oven for 15-20 minutes until the cheese is bubbly and chicken is cooked through.