Melt butter and olive oil in a medium skillet over medium-high heat until butter is melted and foamy.
Reduce heat to medium. Smash garlic with the flat of a knife and cook until lightly browned. Remove garlic and discard.
Add sprouts, cut side down, cover, and cook without stirring on medium-low heat for 10 to 15 minutes or until tender when pierced with a knife. The cut side of the sprouts should get nice and browned, with a nutty, buttery flavor. Top with freshly grated Parmesan cheese and salt and pepper to taste.