Prepare the Chicken: Heat one tablespoon of olive oil over medium-high heat in a large skillet or Dutch oven. Sprinkle salt and pepper on the chicken breasts, then brown either side for 3-4 minutes. Defrost the chicken, brown well, remove from skillet, and set aside.
Caramelize the Onions: Add one tablespoon of olive oil and one tablespoon of butter to the same skillet. Put in the sliced onions when the butter has melted and cook over medium-low heat. Stir frequently, cooking the onions until golden brown and caramelized for about 15 minutes. And if the onions start to stick, pour in a splash of chicken broth to deglaze the pan.
Add Garlic and Rice: Stir in minced garlic and cook one minute longer, until fragrant. Add the rice and cook for 2 minutes, letting it toast and take up the flavor.
Simmer the Dish: Pour the chicken broth into the skillet, scraping up any bits from the bottom of the vessel. Add the thyme, then season with salt and pepper. Return the chicken breasts to the pan, nestling them into the rice mixture. Bring the broth to a simmer, cover, and cook for 20-25 minutes or until the rice is tender and the chicken is cooked.
Finish the Dish: If desired, sprinkle the shredded Gruyère cheese over the chicken and rice during the last 5 minutes of cooking. Cover the pan to melt the cheese.
Serve and Enjoy: Once the dish is ready, serve it hot, garnished with fresh thyme or additional cheese, if desired. Enjoy your delicious French Onion Chicken and Rice!