1. In a large oven-safe skillet over medium-low heat, melt butter and olive oil, then add sliced onions and salt.

2. Cook onions slowly for 30-40 minutes, stirring occasionally, until deeply golden brown and soft.

3. Stir in minced garlic and thyme, cook for 1 minute until fragrant, then remove the onion mixture from the skillet and set aside.

4. Pat chicken dry, season with salt and pepper, and sear in the same skillet over medium-high heat for 4-5 minutes per side until golden brown.

5. Remove chicken from the skillet and set aside.

6. Reduce heat to medium, deglaze the skillet with white wine or broth, scraping up the browned bits from the bottom of the pan.

7. Allow the liquid to reduce by half, then pour in the beef broth, bring to a simmer, and adjust seasoning.

8. Return the seared chicken to the skillet, top each piece with caramelized onions, and simmer for 10-15 minutes until cooked through (165°F).

9. Preheat the broiler, sprinkle shredded Gruyère cheese over the chicken and onions.

10. Broil for 2-3 minutes, watching closely, until the cheese is melted, bubbly, and golden brown.

11. Let the dish rest for a few minutes before serving.