Preheat oven to 425 F. Add 1 tbsp olive oil to a large skillet over med-high heat. Meanwhile, slice chicken in half lengthwise then season with salt and pepper.
Saute chicken for 4-5 minutes each side, then remove from heat and wipe pan clean with a paper towel. Add 1 tsp olive oil to skillet then add onions and cook for 10 minutes, stirring occasionally until onions brown.
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 somewhat browned, mix in flour and then add red wine and beef broth, mixing well to combine.
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!