Season the chicken breasts with salt and pepper. Heat a large oven-safe skillet over medium-high heat and add the olive oil. Sear the chicken breasts on both sides until golden brown, about 3-4 minutes per side. Transfer the chicken to a plate and set it aside.
In the same skillet, melt the butter over medium heat. Add the thinly sliced onions and cook, stirring occasionally, until they become golden brown and caramelized, about 15-20 minutes. This slow cooking process allows the natural sugars in the onions to develop and adds depth of flavor.
Pour the broth into the skillet, scraping the bottom to release any browned bits (this adds flavor). Bring the mixture to a simmer and let it cook for a couple of minutes to allow the flavors to meld together.
Transfer the skillet to the preheated oven and bake for approximately 20-25 minutes, or until the chicken reaches an internal temperature of 165F. The baking time may vary depending on the thickness of the chicken breasts.
Remove the skillet from the oven and sprinkle the shredded Swiss cheese over the chicken and onions. Return the skillet to the oven and broil for an additional 2-3 minutes, or until the cheese is melted and lightly browned.