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. (If at any point the brown bits on the bottom of the pan get too dark, add 1 to 2 tablespoons of beef stock or water to the skillet and scrape up the brown bits.)
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. Serve with slices of baguette, mashed potatoes, or rice.
Notes
Stove-top only method: If you don’t feel like turning on the oven, after topping the chicken with the cheese, place a lid on the skillet. Cook over medium-low heat for 5 to 7 minutes until the cheese is melted.
No meat mallet? No problem. You can use your rolling pin. Note that raw meat and wood cooking implements don’t mix, so if you’re using a wooden rolling pin, I recommend placing the chicken cutlets between pieces of plastic wrap or parchment paper before pounding them.
To easily strip the leaves from a sprig of thyme, hold the top of the stem with one hand, then use your other hand to pinch the stem just below and pull down, stripping all of the leaves away from the main stem. The thinner, pale green parts of the stem are tender, so don’t fret too much about those; we’re just aiming to remove the woody portions.
Add white wine. For added flavor, you can replace up to 1/2 cup of the broth with a dry white wine. This elevates the flavor of the onion sauce and is a trick I love to do when serving this dish at a dinner party!