Prep the Chicken: Cut a horizontal slit in each chicken breast to create a pocket, being careful not to cut all the way through. Season both sides with salt and pepper.

Cook the Onions: In a large skillet over medium heat, melt the butter and olive oil. Add the onions and cook, stirring occasionally, until caramelized and golden brown, about 25-30 minutes. Season with salt and pepper.

Add Garlic and Thyme: Stir in the garlic and thyme and cook for an additional 2 minutes. Remove from heat.

Preheat the Oven: Preheat your oven to 375°F (190°C).

Stuff the Chicken: Stuff each chicken breast with a generous portion of the onion mixture and about 1/4 cup of the shredded cheese. Secure the opening with toothpicks if needed.

Brown the Chicken: In the same skillet over medium-high heat, brown the stuffed chicken breasts on both sides, about 2-3 minutes per side. Remove from the skillet.

Create the Sauce: In the same skillet, add the beef broth, scraping up any browned bits. Simmer for a few minutes until slightly reduced.

Bake the Chicken: Return the chicken breasts to the skillet, spooning some of the sauce over them. Transfer the skillet to the preheated oven and bake for 20-25 minutes, or until the chicken is cooked through.

Add More Cheese: Sprinkle the remaining cheese over the chicken and return to the oven for 2-3 minutes, or until the cheese is melted and bubbly.

Serve: Serve the French Onion Stuffed Chicken hot with a side of mashed potatoes, green beans, or your favorite vegetable.