In a skillet, melt 1 Tbsp. butter and cook onion and broccoli and until onion is softened, 5 to 7 minutes. Add garlic and cook for one minute. Add turkey, season with salt and pepper, and cook until warm. Remove from skillet.
Build sandwiches: broccoli & turkey mixture topped with a piece of cheese.
Melt 2 Tbsp. butter in skillet and place 4 sandwiches directly onto butter. Cook until golden brown. Flip, adding 1 Tbsp. butter more, if needed. Cook until golden brown. Remove and slice to serve!