Season both sides of the chicken breasts with salt and black pepper.
Heat olive oil in a large skillet over medium-high heat until shimmering.
Add the chicken breasts to the skillet and cook for 5-6 minutes on each side until golden brown and cooked through. Remove the chicken from the skillet and set aside.
Reduce heat to medium. Add butter to the same skillet. Once melted, add the sliced onions and sugar.
Cook the onions, stirring frequently, for about 15 minutes or until they are deeply caramelized and soft.
Add the minced garlic and cook for an additional 1 minute until fragrant.
Pour in the white wine to deglaze the pan, scraping up any browned bits from the bottom. Let it simmer for 2 minutes until slightly reduced.
Add the beef broth and fresh thyme leaves. Stir to combine and bring the mixture to a simmer.
Return the chicken breasts to the skillet, spoon some of the onion mixture and broth over the chicken.
Place a slice of Swiss cheese on top of each chicken breast. Cover the skillet with a lid and cook for 3-5 minutes until the cheese is melted and bubbly.
Remove from heat, garnish with additional fresh thyme leaves, and serve immediately.