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 are deep golden and softened, about 20–25 minutes. Stir in balsamic vinegar during the last few minutes if desired, then remove from heat and set aside.
03 - Pat chicken breasts dry with paper towels 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 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 breasts for 2–3 minutes per side until a deep golden crust forms. The chicken will not be fully cooked through at this stage. Transfer seared breasts to the prepared baking sheet.
05 - Spoon a generous portion of caramelized onions over each chicken breast. Evenly distribute shredded mozzarella and cheddar cheese on top of the onions.
06 - Bake in the preheated oven for 15–18 minutes, or until the chicken registers 165°F internally and the cheese is fully melted and bubbly.
07 - Remove from the oven and garnish with freshly chopped parsley. Serve immediately alongside a green salad or roasted vegetables.