Preheat oven to 425 F. Add 1 tbsp olive oil to a large skillet over medium-high heat.
Saute chicken for 4-5 minutes each side, then remove from heat. Add remaining olive oil to skillet, lower heat to medium, then add onions, season with sugar and salt, and cook for 25-30 minutes, stirring occasionally until onions become caramelized.
Meanwhile, toss broccoli with 1 tsp olive oil, garlic, salt and pepper on a parchment-lined sheet pan. Bake for 8-10 minutes until broccoli is tender.
Once onions are softened and browned, mix in cornstarch and then add red wine and beef broth, mixing well to combine. Cook another 2-3 minutes until sauce thickens.
Add chicken back to skillet and cover with onion mixture, then top with cheese and put lid on top of skillet to let cheese melt, about 2 minutes.
Serve chicken with broccoli and enjoy!