Season both sides of the chicken breasts with salt and black pepper.
Heat olive oil in a large skillet over medium-high heat.
Add the chicken breasts to the skillet and cook for 5-6 minutes on each side until golden brown and cooked through. Remove chicken from the skillet and set aside.
Reduce heat to medium and add butter to the same skillet.
Add the sliced onions and cook, stirring frequently, for about 15 minutes until the onions are deeply caramelized.
Add the minced garlic and fresh thyme leaves to the onions and cook for 1 minute until fragrant.
Sprinkle the flour over the onions and stir well to combine. Cook for 1-2 minutes to remove the raw flour taste.
Slowly pour in the white wine while stirring to deglaze the pan, scraping up any browned bits.
Add the beef broth and Worcestershire sauce, stirring to combine. Bring the mixture to a simmer and cook for 5 minutes until the sauce thickens slightly.
Return the chicken breasts to the skillet, nestling them into the onion mixture.
Place a slice of Swiss cheese on top of each chicken breast.
Cover the skillet with a lid and cook for an additional 3-5 minutes until the cheese is melted and bubbly.
Remove from heat and serve the chicken topped with the caramelized onion sauce.