Pat the chicken cutlets completely dry with paper towels. Season both sides evenly with kosher salt, black pepper, and garlic powder, then lightly dredge through flour, shaking off any excess. In a large 12-inch oven-safe skillet over medium-high heat, melt 1 tablespoon of butter with 1 tablespoon of olive oil. Sear the chicken for 4 to 5 minutes per side until golden brown and cooked through (165°F / 74°C internal temp). Transfer to a warm plate and set aside.

Reduce heat to medium. Melt the remaining 2 tablespoons of butter in the same skillet. Add the sliced yellow onions with a pinch of salt. Cook, stirring occasionally, for 15 to 18 minutes until the onions turn deeply golden-brown, sweet, and jammy. If the onions begin browning too quickly, lower the heat to medium-low and add a splash of broth.

Add the minced garlic and fresh chopped thyme to the caramelized onions. Sauté for 1 minute until fragrant. Pour in the beef broth, Worcestershire sauce, balsamic vinegar, and Dijon mustard, using a wooden spoon to scrape up all the flavorful browned bits (fond) stuck to the bottom of the skillet. Bring to a gentle simmer for 3 minutes.

Lower heat to medium-low and stir in the heavy whipping cream until velvety and well combined. Simmer for 2 minutes until slightly thickened. Nestle the seared chicken cutlets back into the pan, spooning the luscious caramelized onions and creamy sauce over each cutlet.

Generously top the chicken and sauce with the shredded Gruyère and mozzarella cheese. Transfer the skillet under your oven's high broiler for 3 to 4 minutes until the cheese is melted, bubbling, and beautifully blistered with golden spots. Garnish with fresh thyme sprigs and serve immediately.