Heat 2 tablespoons olive oil in a large oven-safe skillet over medium heat. Add the sliced onions and stir to coat with the oil. Cook for 10-12 minutes until softened.
Stir in thyme, salt, and pepper. Add ¼ cup of chicken broth. Cook for another 12-14 minutes until golden brown. Stir in balsamic vinegar. Transfer the onions to a bowl or plate and set aside.
Preheat the oven to 375 degrees.
Heat another tablespoon of olive oil in the same skillet.
Season the chicken on both sides with paprika, thyme, salt, and pepper. Add the chicken to the skillet and cook for 3-4 minutes until brown. Flip and cook for another 3-4 minutes.
Add the caramelized onions back to the skillet along with ¼ cup of chicken broth.
Transfer the skillet to the oven. Bake for 8-10 minutes until the internal temperature reaches 165 degrees.
Top the chicken with the shredded cheese and bake for another 5 minutes until the cheese is melted. Serve with rice, pasta, or mashed potatoes.
Use freshly shredded cheese for the best melting quality.
Use an instant-read thermometer to check the doneness of the chicken. The chicken needs to reach a safe temperature of 165 degrees. The length of cooking time will depend on the size and thickness of the chicken breasts.
If using chicken thighs, be sure to cook to a safe internal temperature. While chicken thighs are safe at 165 degrees, they will be more tender at 175-195 degrees.
To make ahead, take care to not overcook the chicken since it will be reheated. Wait until you reheat the chicken to top it with the cheese.