Cut through the chicken breasts lengthwise to create 4 evenly sized cutlets. If necessary, use a meat mallet to pound them to an even thickness, about ¾ inch thick. Season the cutlets all over with ½ teaspoon of salt and pepper.

Add the flour to a shallow bowl. Dredge the chicken cutlets in the flour, shaking off any excess. Reserve 2 tablespoons of the remaining flour.

Meanwhile, heat a large, 12-inch oven-proof skillet over medium-high heat. Add the olive oil, then the chicken in a single layer. Cook for 3 minutes per side until lightly golden. (At this point, the chicken may not be cooked through.) Remove the pan from the heat and place the chicken on a plate. Set aside.

Preheat the oven to 400°F.

Return the skillet to medium-high heat. Melt the butter, then add the onions and the remaining ½ teaspoon of salt. Cook, stirring frequently, for 10 minutes. Reduce the heat to medium and cook for another 10 minutes, stirring occasionally, until the onions are golden brown.

Add the garlic and sprinkle the reserved 2 tablespoons of flour over top. Stir well to combine and cook for 2 more minutes.

Add the beef broth, Worcestershire sauce and fresh thyme leaves. Stir to combine, scraping the brown bits from the bottom of the pan. Increase the heat to medium to bring the mixture to a simmer for 2 minutes.

Nestle the chicken cutlets into the onion mixture. Top each piece of chicken with 1 to 2 tablespoons of the onion mixture, then the grated cheese.

Place it in the oven and bake for 6 to 8 minutes until the cheese is melted. Sprinkle with fresh thyme leaves and more black pepper before serving.