01 - Preheat oven to 400°F. Line a baking sheet with parchment paper and set aside.
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 20-25 minutes. Stir in balsamic vinegar during the last few minutes if desired. Remove from heat and set aside.
03 - Pat chicken breasts dry with paper towels. Season both sides generously with salt, pepper, garlic powder, and paprika. Set up a breading station: dredge each breast in flour, dip into beaten eggs, then coat thoroughly with a mixture of panko breadcrumbs and grated Parmesan cheese.
04 - Heat olive oil in a large skillet over medium-high heat. Sear breaded chicken breasts for 2-3 minutes per side until a golden crust forms. Do not cook through. Transfer seared chicken 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 portion.
06 - Bake for 15-18 minutes, or until the chicken reaches an internal temperature of 165°F and the cheese is fully melted and bubbly.
07 - Remove from oven and garnish with chopped fresh parsley. Serve immediately with a green salad or roasted vegetables.