Pat the chicken cutlets dry and season both sides with garlic powder, Italian herbs, salt, and pepper. Heat 1 tablespoon olive oil and 1 tablespoon butter in a 12-inch oven-safe skillet over medium-high heat. Sear the chicken for 3–4 minutes per side until golden brown. Transfer to a plate.

Melt the remaining 2 tablespoons butter in the same skillet. Add the thinly sliced onions, 1/2 teaspoon sugar, and a tiny pinch of baking soda. Cover with a lid for 4 minutes to soften, then uncover and sauté over medium-high heat for 7–8 minutes, stirring occasionally, until jammy and golden brown.

Stir the minced garlic, fresh thyme, and flour into the caramelized onions and cook for 60 seconds. Slowly pour in the beef broth and balsamic vinegar, whisking until a glossy, spoon-coating onion gravy forms.

Nestle the seared chicken cutlets back into the skillet and spoon half of the caramelized onion gravy over top. Lay 1 slice of provolone and a generous mound of shredded Gruyère over each cutlet. Broil on HIGH for 2–3 minutes until bubbly and blistered golden brown.