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 sliced onions, sugar, and salt. Cook, stirring occasionally, until onions are deep golden and soft, about 20 to 25 minutes. Stir in balsamic vinegar during the last few minutes if using. 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 in beaten eggs, then coat thoroughly with a mixture of panko breadcrumbs and grated Parmesan.
04 - Heat olive oil in a large skillet over medium-high heat. Brown chicken for 2 to 3 minutes per side until golden. Do not cook through. Transfer chicken to the prepared baking sheet.
05 - Top each chicken breast with a generous spoonful of caramelized onions, then sprinkle evenly with shredded mozzarella and cheddar cheese.
06 - Bake for 15 to 18 minutes, or until the chicken is cooked through and the cheese is melted and bubbly.
07 - Garnish with chopped fresh parsley before serving.