Pat chicken cutlets dry with paper towels. In a shallow dish, whisk together flour, garlic powder, 1/2 tsp salt, and black pepper. Dredge each chicken cutlet lightly in the flour mixture, shaking off any excess.

Heat 2 tablespoons of olive oil in a large 12-inch oven-safe cast-iron skillet over medium-high heat. Sear the cutlets for 4 to 5 minutes per side until golden brown and almost cooked through. Transfer to a plate and keep warm.

Reduce the skillet heat to medium. Melt the butter in the same pan, then add all the sliced onions and remaining 1/2 tsp salt. Cook, stirring occasionally, for 15 to 18 minutes until the onions are deeply browned, soft, and sweet.

Stir in minced garlic, fresh thyme leaves, and balsamic vinegar. Cook for 1 to 2 minutes until aromatic and the pan bottom is deglazed.

Pour in the beef broth and Dijon mustard, scraping up any delicious browned bits from the bottom of the skillet. Bring to a simmer and let reduce by half for 3 to 4 minutes.

Stir in the heavy cream, allowing the sauce to bubble gently into a rich, velvety golden-brown pan sauce.

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

Blanket the top with shredded Gruyère and Swiss cheese. Transfer the skillet to the oven and broil on high for 3 to 4 minutes until the cheese is bubbling, gooey, and spotted with golden-brown blisters.

Garnish with fresh thyme leaves and cracked black pepper. Let rest 5 minutes, then serve hot straight from the skillet!