In a large deep-sided skillet, melt the butter over medium-low heat. Add the sliced onions to cook slowly until they're caramelized, stirring occasionally. If they start browning too quickly, turn down the heat a little more. This process should take 20 - 25 minutes.
Add the thyme, oregano, and garlic to the skillet. Stir into the onions and cook for another 3 minutes. Then remove the onions to a bowl and cover to keep warm.
Wipe out the skillet, place it over medium heat, and pour in the olive oil. Season the chicken with salt and pepper on both sides and place the thighs in the hot oil. Cook each side for about 5 minutes until browned, and then transfer the chicken to a plate.
Preheat your oven to 375 degrees F.
Place the onions back into the skillet and pour in the white wine and beef broth. Nestle the chicken in the onion mixture and cook for 20 minutes until the chicken is cooked through (internal temp 165 degrees F) and the sauce has reduced by half.
Top the chicken with the mozzarella and Swiss cheese and bake for 5 to 10 minutes until the cheese is melted and bubbly. (If you want the cheese to have those golden brown bubbles, broil it for 1-2 minutes once melted.)
Remove the skillet from the oven and sprinkle with chopped parsley. Serve warm with your favorite sides - we highly recommend mashed potatoes.
Notes
Store leftovers in an airtight container in the refrigerator for up to 5 days.
If you prefer to use bone-in thighs, you'll need to add 10-15 minutes to the cooking time in the oven. Use a meat thermometer to check that the chicken is cooked through.
If you don't have an oven-safe skillet, pour the onion mixture into a casserole dish, place the chicken breasts over the onion, and then top with cheese. Bake as directed.