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 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, and 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 2–3 minutes per side until a golden crust forms. The chicken should not be fully cooked at this stage. Transfer seared breasts to the prepared baking sheet.
05 - Spoon a generous portion of caramelized onions over each chicken breast. Top evenly with shredded mozzarella and cheddar cheeses.
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 - Sprinkle chopped fresh parsley over the finished chicken. Serve immediately with a simple green salad or roasted vegetables.