Preheat your oven to 350ºF (175ºC).

In a large, oven-proof pan or skillet, heat 2 tablespoons of olive oil over medium heat. Add the sliced onions, sprinkle with salt, and cook for 15-20 minutes, stirring occasionally, until they become softened and caramelized.

Add the balsamic vinegar to the skillet and cook for an additional 3-5 minutes, allowing the flavors to deepen.

Remove the caramelized onions from heat and transfer them to a medium bowl.

Using the same skillet, heat the remaining olive oil over medium-high heat.

Season the chicken breasts or thighs with salt, pepper, thyme, and sage. Place them in the skillet and sear on both sides until golden brown.

Once the chicken is seared, remove it from heat and set it aside.

Pour the beef broth into the skillet, bringing the mixture to a boil. Use a spatula to scrape up any browned bits from the bottom of the pan, as they add incredible flavor.

Reduce the heat to medium-low and whisk in the flour, stirring until the mixture becomes smooth. Cook for 5-7 minutes, or until the gravy has thickened.

Taste the gravy and adjust the seasoning with salt and pepper, if needed.

Return the chicken and caramelized onions to the skillet, stirring them together with the beef gravy. Ensure that the chicken is evenly coated.

Sprinkle the grated Swiss cheese over the top of the skillet, creating a luscious layer.

Transfer the skillet to the preheated oven and cook until the cheese is melted and bubbly, and the chicken is fully cooked.

Remove the skillet from the oven, and with great anticipation, serve this mouthwatering French Onion Chicken Skillet hot. Savor each bite and revel in the harmonious blend of flavors!