To have the chicken breasts cooked evenly, we need to pound them thin to the same thickness. Cover them with plastic wrap and pound them to ¾-inch thick.
Season the chicken with salt, pepper, and dried thyme.
Heat 1 ½ tbsp olive oil in a large skillet over medium-high heat and brown the chicken breasts for about 5 minutes on each side. The skillet has to be very hot before placing the chicken in it.
Transfer the browned chicken breasts to a plate and wipe the skillet clean with a paper towel.
Heat the skillet over medium heat and melt the butter along with the remaining 1 ½ tbsp olive oil. Add the sliced onions and cook them for about 20 minutes or until golden brown and caramelized. Add the minced garlic and cook 30 seconds more.
Mix in the flour until well combined and pour in the beef broth and Worcestershire sauce. Mix well.
Return the chicken breasts to the skillet, place two fresh thyme sprigs on top and simmer for about 20 minutes or until the sauce is thick and the chicken is cooked through.
Top each breast with two slices of mozzarella cheese, place the lid on top and cook until the cheese has melted.