Pat the chicken cutlets dry with paper towels. In a shallow dish, whisk together flour, garlic powder, Italian seasoning, salt, and black pepper. Lightly dredge each cutlet, shaking off any excess.

Heat 1 tablespoon olive oil and 1 tablespoon butter in a large oven-safe cast iron skillet over medium-high heat. Sear the chicken for 4 to 5 minutes per side until golden brown and cooked to 160°F (71°C). Transfer chicken to a warm plate and set aside.

Reduce the heat to medium. Add the remaining 1 tablespoon of olive oil and 1 tablespoon of butter into the same skillet. Add the sliced onions and a pinch of salt.

Cook, stirring occasionally, for 15 to 20 minutes until the onions soften, collapse, and turn a rich, jammy deep golden-brown. Stir in the minced garlic and fresh thyme, cooking for 1 minute until fragrant.

Pour in the balsamic vinegar and beef broth, using a wooden spoon to scrape up all the delicious browned fond stuck to the bottom of the skillet. Bring to a gentle simmer for 3 minutes until slightly reduced.

Stir in the heavy cream and let simmer for another 2 minutes until a luxurious, velvety French onion pan gravy forms.

Nestle the seared chicken cutlets back into the simmering skillet, spooning generous heaps of caramelized onions and sauce over each breast.

Top each cutlet generously with a blanket of shredded Gruyère and mozzarella cheese. Transfer skillet into the oven on BROIL (HIGH) for 3 to 4 minutes until the cheese is bubbling vigorously and blistered with golden-brown spots.

Garnish with fresh thyme leaves and cracked black pepper. Serve piping hot with creamy garlic mashed potatoes or crusty French bread for dipping!