In a 10-inch cast-iron skillet over medium heat, combine the olive oil, butter, and sliced onions. Saute, stirring occasionally, until the onions become soft (approximately 10 minutes).
Sprinkle 1 teaspoon granulated sugar and ½ teaspoon coarse sea salt over the onions. Add fresh thyme and a bay leaf to the pan. Continue sautéing over medium heat to medium-low heat until the onions become deep golden brown. Stir every few minutes to prevent onions from burning. If the color becomes too dark on some of the onions while still cooking, add a splash of white wine or beef broth to the pan and stir. This process of caramelizing the onions should take about 15 to 20 minutes longer after adding the sugar, salt, thyme, and bay leaf. In the final minute or two of cooking, add minced garlic and saute for just a couple of minutes; do not burn the garlic.
While the onions are cooking, prepare the chicken. Remove from packaging and pat with a paper towel to make sure they are completely dry. Use a meat mallet to pound out chicken to an even ½ to ¾-inch thick. Sprinkle with pepper on both sides.
Remove caramelized onions from the skillet using a slotted spoon, keeping as much butter/oil in the pan as possible.
Without wiping out the skillet, turn the heat to medium to medium-high and add the chicken. Sear on each side until browned, about 3 to 4 minutes on each side.
Preheat oven to 350°F.
Remove the chicken breasts from the skillet and set them aside momentarily.
Place the onions back into the skillet and return to medium heat. Sprinkle with flour and sauté for 1 to 2 minutes to cook the flour.
Add the wine to the onion mixture and simmer/stir for 2 to 3 minutes while the alcohol evaporates and the browned bits on the bottom of the pan loosen. If you are not using wine, proceed with step 10.
Add the beef broth to the onions and bring the mixture to a simmer. Simmer for 2 to 3 minutes until the mixture thickens slightly and a deep brown gravy forms.
Nestle the browned chicken breasts down into the onion gravy and spoon some of the sauce up over the chicken. Top each chicken breast with a ¼ of each of the cheeses.
Transfer the skillet to the oven and cook for 20 to 25 minutes until the cheese has melted and the meat registers an internal temperature of 165°F.
Remove from oven. Spoon onions and sauce over the chicken and serve. Garnish with fresh thyme.