Season & Sear Chicken: Pat chicken cutlets dry with paper towels. Season both sides evenly with garlic powder, dried oregano, kosher salt, and freshly cracked black pepper. Heat 1 tablespoon olive oil and 1 tablespoon butter in a 12-inch oven-safe skillet over medium-high heat. Sear cutlets for 4 to 5 minutes per side until golden brown and cooked to 165°F (74°C). Transfer chicken to a plate.
Caramelize Onions: In the same skillet over medium heat, melt the remaining 2 tablespoons butter. Add thinly sliced onions and 1/4 teaspoon salt. Cook, stirring frequently with a wooden spoon, for 12 to 14 minutes until onions are soft, deep golden-brown, and sweet. Add minced garlic and fresh thyme leaves; sauté for 60 seconds until fragrant.
Simmer Gravy: Pour in beef bone broth and balsamic vinegar. Scrape up the flavorful browned fond from the skillet bottom. Simmer for 3 to 4 minutes until the broth reduces slightly into a glossy, rich onion sauce.
Add Chicken & Cheese: Nestle seared chicken cutlets and any accumulated juices back into the skillet. Spoon caramelized onions over each chicken piece, then top generously with shredded Gruyère and Swiss cheeses.
Broil & Serve: Transfer skillet to oven and broil on HIGH for 3 to 4 minutes until cheese is completely melted, bubbly, and spotted with golden blisters. Garnish with fresh thyme sprigs and serve hot with toasted baguette slices for dipping.