Sear the Chicken: In a large skillet, heat 1 tablespoon of olive oil over medium-high heat. Add the chicken breasts and cook for about 5-6 minutes on each side until they are golden brown and cooked through. Remove the chicken from the skillet and set aside.

Caramelize the Onions: In the same skillet, add the remaining tablespoon of olive oil. Add the sliced onions and cook over medium heat, stirring occasionally, for about 15-20 minutes until they are caramelized and soft.

Add Flavor: Once the onions are caramelized, stir in the thyme and balsamic vinegar. Cook for another minute until the vinegar has evaporated.

Make the Sauce: Pour in the beef broth and stir well, scraping the bottom of the pan to deglaze it. If you want a thicker sauce, whisk in the flour at this stage. Let the mixture simmer for about 5 minutes to allow the flavors to meld.

Combine Chicken and Sauce: Return the seared chicken to the skillet, spooning the onion mixture over the top. Sprinkle the shredded cheese evenly over the chicken.

Melt the Cheese: Cover the skillet and reduce the heat to low. Let it cook for an additional 5-7 minutes until the cheese is melted and bubbly.

Garnish and Serve: Remove from heat, garnish with fresh parsley, and serve warm.