2 pounds Vidalia onions, sliced
2 pounds boneless, skinless chicken breasts or thighs
2 cups beef broth
1 cup swiss cheese, grated
1/3 cup extra-virgin olive oil, divided
2 tablespoons all-purpose flour
2 teaspoons balsamic vinegar
1/2 teaspoon dried thyme
1/2 teaspoon dried sage
Kosher salt and freshly ground pepper, to taste
Preheat oven to 350 F.
Heat 2 tablespoons olive oil in a large, oven-proof pan or skillet over medium heat. Season with salt and cook, stirring occasionally, for 15-20 minutes, or until softened and caramelized.
Add balsamic vinegar and cook for another 3-5 minutes or until darkened, but not burned.
Remove onions from heat and transfer to a medium bowl.
Heat remaining olive oil in skillet and raise heat to medium-high.
Season chicken breasts with salt, pepper, thyme and sage, then place in skillet and sear on both sides until golden brown. Remove chicken from heat and set aside.
Pour beef broth into skillet and bring mixture to a boil, scraping up stuck bits from the bottom of pan.
Return heat to medium-low and whisk in flour, stirring until smooth. Cook for 5-7 minutes, or until mixture has thickened. Taste and adjust seasoning, if necessary.
Return chicken and onions to pan, stir together with beef gravy, then top with grated swiss cheese.
Transfer skillet to oven and cook until cheese is melted and bubbly, and chicken is cooked through.
Remove from oven and serve hot.