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