Caramelize the Onions:

Heat the olive oil and butter in a skillet over medium heat. Add the sliced onions, thyme, salt, and pepper. Cook, stirring occasionally, until the onions are golden and caramelized, about 15-20 minutes.

Add the beef broth to deglaze the pan, scraping up any browned bits, and simmer for 2-3 minutes. If you prefer a thicker onion mixture, stir in the flour and cook for 1-2 more minutes. Set aside.

Prepare the Chicken:

Preheat your oven to 375°F (190°C). Using a sharp knife, cut a pocket into the side of each chicken breast, being careful not to cut all the way through. Season the chicken with salt and pepper.

Stuff the Chicken:

Stuff each chicken breast with a generous portion of the caramelized onion mixture and a mix of the Gruyère and mozzarella cheeses. Reserve some onions and cheese for topping. Use toothpicks to secure the openings if needed.

Sear the Chicken:

Heat a large oven-safe skillet over medium-high heat with a drizzle of olive oil. Sear the stuffed chicken breasts for 2-3 minutes per side until golden brown.

Bake the Chicken:

Spoon the reserved onions on top of the chicken and sprinkle with the remaining cheese mixture. Transfer the skillet to the preheated oven and bake for 15-20 minutes, or until the chicken reaches an internal temperature of 165°F (74°C).

Serve:

Remove the toothpicks (if used) and garnish with fresh parsley before serving.