Pat chicken cutlets dry with paper towels. Season both sides evenly with 1/2 teaspoon salt and 1/4 teaspoon black pepper.

Heat 1 tablespoon olive oil and 1 tablespoon butter in a large oven-safe cast iron skillet over medium-high heat. Sear chicken cutlets for 3 to 4 minutes per side until deeply golden. Transfer chicken to a plate (it will finish cooking later).

Lower heat to medium. Add remaining 2 tablespoons butter and sliced onions to the skillet. Cook slowly for 12 to 15 minutes, stirring frequently with a wooden spatula, until deep golden-brown, sweet, and jammy.

Add minced garlic, fresh thyme leaves, and flour to the caramelized onions. Stir constantly for 60 seconds to toast the flour.

Pour in the rich beef bone broth to deglaze the skillet, scraping up every bit of savory caramelized fond from the bottom. Simmer for 2 minutes until the gravy thickens into a glossy, velvety glaze.

Nestle the seared chicken cutlets back into the skillet, spooning the luscious caramelized onions and gravy over the top. Sprinkle shredded Gruyère and mozzarella generously over each cutlet.

Broil on high in the oven for 3 to 4 minutes until the cheese is completely melted, bubbly, and speckled with golden-brown spots. Garnish with fresh thyme and serve immediately.