Season chicken cutlets generously on both sides with kosher salt, black pepper, and dried oregano.
Heat olive oil and 1 tablespoon butter in a heavy 12-inch cast-iron skillet over medium-high heat. Sear chicken cutlets for 4 to 5 minutes per side until beautifully golden brown and cooked through (internal temperature 165u00b0F / 74u00b0C). Transfer chicken to a clean plate and tent loosely with foil.
Reduce skillet heat to medium. Add remaining 2 tablespoons butter to the pan drippings. Add the sliced yellow onions, a pinch of salt, and fresh thyme leaves. Cook slowly for 18 to 20 minutes, stirring frequently, until onions turn soft, jammy, and deeply caramelized golden brown.
Stir in minced garlic and cook for 1 minute until fragrant. Sprinkle all-purpose flour over the onions and cook for 1 minute, stirring constantly to coat.
Slowly pour in the rich beef broth and Worcestershire sauce, using a wooden spatula to scrape up all the savory browned fond from the bottom of the skillet. Simmer for 2 to 3 minutes until the onion gravy thickens into a glossy sauce.
Nestle the seared chicken cutlets back into the pan, spooning generous mounds of caramelized onions over each piece. Top evenly with shredded Gruyu00e8re and mozzarella cheese.
Cover the skillet with a lid or transfer directly under a preheated oven broiler for 2 to 3 minutes until the cheese is bubbling, gooey, and toasted with golden brown spots. Garnish with fresh thyme sprigs and chopped parsley. Serve hot directly from the skillet!