01 - Preheat oven to 400°F. Line a baking sheet with parchment paper.
02 - In a large skillet, heat butter and olive oil over medium-low heat. Add the thinly sliced onions, sugar, and salt. Cook, stirring occasionally, until the onions are deep golden and softened, about 20 to 25 minutes. Stir in balsamic vinegar during the last few minutes if using. Set aside.
03 - Pat the chicken breasts dry with paper towels. Season both sides with salt, pepper, garlic powder, and paprika. Set up a breading station: dredge each breast in flour, dip in beaten eggs, then coat thoroughly with the panko and Parmesan mixture.
04 - Heat olive oil in a large skillet over medium-high heat. Sear the breaded chicken for 2 to 3 minutes per side until golden brown. The chicken will not be fully cooked at this stage. Transfer the seared breasts to the prepared baking sheet.
05 - Top each chicken breast with a generous spoonful of caramelized onions. Sprinkle an even layer of mozzarella and cheddar cheese over each piece.
06 - Bake for 15 to 18 minutes, or until the chicken is fully cooked through and the cheese is melted and bubbly.
07 - Garnish with chopped fresh parsley before serving.