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 and cook for 5 to 6 minutes on each side until golden brown and cooked through. Remove chicken from skillet and set aside.
Reduce heat to medium. Add butter to the same skillet. Once melted, add the sliced onions and cook, stirring occasionally, until caramelized and golden brown, about 15 minutes.
Add the minced garlic and fresh thyme leaves to the onions and cook for 1 minute until fragrant.
Pour in the white wine and scrape the bottom of the skillet to deglaze, cooking until the wine reduces by half, about 3 minutes.
Add the beef broth and Worcestershire sauce. Stir and bring the mixture to a simmer.
Return the chicken breasts to the skillet, nestling them into the onion mixture. Spoon some onions over the chicken.
Reduce heat to low, cover the skillet, and simmer for 5 minutes to allow flavors to meld.
Uncover the skillet and place one slice of Swiss cheese on each chicken breast. Cover again and cook until the cheese melts, about 2 to 3 minutes.
Sprinkle chopped fresh parsley over the skillet before serving.