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 sliced onions, sugar, and salt. Cook, stirring occasionally, until the onions are deep golden and soft, about 20-25 minutes. Stir in balsamic vinegar during the last few minutes if desired. Set aside.
03 - Pat the chicken breasts dry and season both sides with salt, pepper, garlic powder, and paprika. 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 chicken for 2-3 minutes per side until golden brown. Do not cook through. Transfer the browned chicken to the prepared baking sheet.
05 - Top each chicken breast with a generous spoonful of caramelized onions, then sprinkle evenly with mozzarella and cheddar cheese.
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.