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