In a large oven safe skillet, melt the butter over medium heat. Add the onions and season with the sugar and ½ teaspoon of the salt. Sauté for about 20-25 minutes, stirring occasionally. The onions should be caramelized and brown. Preheat the oven to 400 degrees.
Add the garlic and Italian seasoning. Saute for 2-3 minutes or until fragrant.
Deglaze the pan with the white wine, scraping any brown bits off the bottom. Reduce the wine until there is no more liquid left.
Remove the onions set aside. Add the olive oil to the skillet and increase the heat to medium high.
While the oil heats, season the chicken on both sides with the remaining salt and pepper.
Place the chicken in the hot oil. Sear for 6-7 minutes, without moving the chicken.
Flip and cook for another 6-7 minutes, or until brown. The chicken should be mostly cooked through. Turn the heat to low.
In a medium bowl, whisk together the stock, worcestershire, and corn starch.
Add the stock mixture into the skillet. (If you do not have an oven safe dish, transfer the chicken to a baking dish first. Then add stock, onions, and cheese.)
Top each chicken breast with the onions.
Sprinkle the cheese on top of each chicken breast.
Bake for 15 minutes or until the chicken has reached an internal temperature of 165 degrees. If the chicken is not done, cover and continue cooking.