Heat butter in a large cast-iron skillet over medium until melted. Stir in onions, 1 teaspoon of the thyme, and 1/4 teaspoon of the pepper. Cook, stirring occasionally, until onions are translucent, about 3 minutes. Stir in 1/2 cup of the broth and bring to a boil over medium high. Reduce heat to medium-low; cover and cook, stirring occasionally, until onions are soft, about 18 minutes. Sprinkle with sugar and continue to cook onions over medium-high heat, uncovered, stirring often, until lightly browned around edges, 4 to 5 minutes. Transfer onions to a plate, rinse and wipe skillet clean.

Season chicken cutlets with onion powder, salt, remaining 1 teaspoon thyme, and remaining 1/4 teaspoon pepper. Place flour in a shallow dish. Dredge chicken cutlets one at a time in flour, shaking off excess. Place on a plate.

Heat oil in skillet over medium. Working in 2 batches, add  2 chicken cutlets; cook, turning once halfway through, until golden brown, 5 to 6 minutes total per batch. Transfer to a  plate. Repeat process with remaining 2 cutlets.

Preheat oven to broil with rack 5 inches from heat source. Add sherry to skillet, scraping up any browned bits. Simmer until slightly reduced, about 2 minutes. Stir remaining 1 1/4 cups broth and bouillon until combined. Cook, stirring often, until sauce is slightly thickened, about 4 minutes.

Place chicken cutlets in sauce, top each cutlet with 1/2 cup of the caramelized onions, 1/4 cup of the gruyere and 1 tablespoon of the parmesan. Broil in preheated oven until cheese is melted and lightly golden brown, 3 to 4 minutes. Garnish with black pepper and fresh thyme leaves. Serve with crusty bread.