01 - Preheat oven to 400°F. Line a baking sheet with parchment paper.
02 - In a large skillet, melt butter with olive oil over medium-low heat. Add thinly sliced onions, sugar, and salt. Cook, stirring occasionally, until onions turn deep golden and become soft, about 20 to 25 minutes. Stir in balsamic vinegar during the last few minutes if using, then set aside.
03 - Pat chicken breasts dry and season both sides with salt, pepper, garlic powder, and paprika. Set up a breading station: dredge each breast in flour, dip into beaten eggs, then press into a mixture of panko breadcrumbs and grated Parmesan until evenly coated.
04 - Heat olive oil in a large skillet over medium-high heat. Sear breaded chicken for 2 to 3 minutes per side until a golden crust forms. Do not cook through. Transfer seared chicken to the prepared baking sheet.
05 - Spoon a generous portion of caramelized onions over each chicken breast. Sprinkle shredded mozzarella and cheddar cheese evenly on top.
06 - Bake for 15 to 18 minutes, or until the chicken is fully cooked through and the cheese is melted and bubbly.
07 - Sprinkle chopped fresh parsley over the top before serving. Pair with a green salad or roasted vegetables.