Sear the Chicken Cutlets: Pat chicken cutlets dry with paper towels and season both sides with sea salt, garlic powder, and black pepper. Heat 1.5 tablespoons olive oil in a large 12-inch oven-safe cast iron or stainless steel skillet over medium-high heat. Add chicken cutlets and sear for 4 to 5 minutes per side until deeply golden and cooked to 160°F (71°C). Transfer chicken to a clean plate.

Caramelize the Onions: Reduce skillet heat to medium. Melt the butter in the remaining pan drippings. Add the thinly sliced yellow onions and a pinch of salt. Cook, stirring occasionally, for 20 to 25 minutes. If onions begin to scorch, reduce heat to medium-low and splash with 1 tablespoon water. The onions will break down into sweet, jammy, deep golden-amber ribbons.

Aromatize & Build Gravy: Stir in minced garlic and fresh thyme; cook for 60 seconds until fragrant. Sprinkle 1 tablespoon flour over the onions and cook for 1 minute to remove raw flour taste. Pour in the beef bone broth, balsamic vinegar, and Worcestershire sauce, scraping up all the savory browned bits fond from the bottom of the pan.

Simmer the Sauce: Bring the sauce to a gentle simmer for 3 to 4 minutes until thickened into a glossy, rich French onion pan gravy. Taste and adjust seasoning with extra black pepper or salt if needed.

Layer & Broil: Return the seared chicken cutlets (and any accumulated juices) into the skillet, spooning generous heaps of jammy caramelized onions over each cutlet. Top each chicken piece with a generous mound of shredded Gruyere and mozzarella cheese.

Melt & Serve: Transfer skillet under the oven broiler on high for 3 to 4 minutes (or cover with a lid over low heat on stovetop) until the cheese is completely melted, bubbling, and blistered with golden-brown spots. Garnish with fresh thyme and parsley, and serve immediately with toasted French baguette slices.