sliced and toasted French bread slices to sop up the sauce
Place 2 tablespoons olive oil in pan and heat to medium-low. Add onions and cook until the onions are a deep rich brown color. About 45 minutes. DO NOT TURN THE HEAT TO HIGH to cook it faster. You may need to add a tablespoon of broth occasionally if you see the onions sticking. The onions will reduce down a lot. Don't worry. Its what is supposed to happen.
While the onions are cooking in another pan you can cook the chicken. Turn heat up to high. Add 1 tablespoon of olive oil. Season the chicken with salt and pepper than brown the each side of the chicken for 5 mins per side. Don't overcrowd the pan. Its better to do it in batches and give the chicken room to brown. Remove all chicken from the pan and wait until onions are completed. You can deglaze the pan with wine and then add this mixture into onions when you add the rest of the broth. (turn heat off pan after deglazing until you are ready for it.) What is deglazing: (pour in wine, and scrape up all the browned flavorful bits)
Once the onions are nice and browned, add garlic, thyme and flour to onions and stir vigorously to combine. Continue to cook for 3 minutes. Then add half of the broth and stir and cook for 5 minutes. Then scoop out 1/3 of the onions and place on a plate. Add the rest of the broth, balsamic vinegar, mustard and Worcestershire sauce and bring to a boil.(And the deglazed juices from cooking the chicken) The sauce should start to thicken in a few minutes.
Place chicken back into the pan with thickened onions/broth and then divide the onions you have set aside and place them atop of each piece of chicken. Place into a 350° oven and bake for 20 minutes. Then remove from the oven and turn the oven to broil. Sprinkle the grated cheese over each of the chicken topped with onions. Place pan back in the oven and broil until cheese is melted and golden brown. (about 5 minutes depending on your oven)
Serve Immediately. Serving suggestions: serve over rice, pasta, gnocchi. Serve along French bread slices to soak up the sauce.