In a large oven-safe skillet over medium heat, melt 3 Tablespoons butter. Add onions and cook, stirring occasionally, for about 20-25 minutes.

Add 2 Tablespoons beef broth at 20 minutes, or when needed if onions are sticking, and continue cooking onions down until browned and extremely tender, another 10-15 minutes.

Remove onions from the skillet, and set aside. But do not clean out the pan.

While onions cook: mix salt and pepper, rosemary, sage, and thyme together in a small bowl. Then prepare your chicken by patting it dry, and seasoning it with the salt and pepper herb mixture.

Once onions have cooked and are removed from the skillet, add a Tablespoon of butter to the skillet, and increase heat to medium high. Cook chicken for 4-5 minutes per side until browned on both sides, but not necessarily cooked through.

Transfer chicken to a plate and cover to keep warm.

Preheat oven to 400 degrees F.

Add a little beef broth or the white wine to the skillet to deglaze it, using a wooden spoon to scrape the brown bits off.

Return the onions to the skillet.

Sprinkle flour over the onions and cook while continuously stirring for 1 minute over medium-high heat.

Add remaining beef broth, balsamic vinegar, dijon mustard, and worcestershire sauce to the skillet and continue to cook, stirring continuously, until mixture comes to a boil.

Taste and season with salt and pepper as needed.

Return chicken to skillet and spoon a little sauce over each piece of chicken.

Top each pieces of chicken with 1/4 of the gruyere cheese.

Transfer skillet to your preheated oven and cook for 10 minutes, until chicken is cooked through (internal temp of 165 degrees) and cheese is melted.

Spoon onions and sauce over the top of the chicken, garnish with thyme or parsley and black pepper, and serve immediately.