Heat the oven to 425°F (220°C) and lightly grease a rimmed baking sheet.
Warm the olive oil in a large skillet over medium-high heat.
Arrange the halved Brussels sprouts cut side down in one layer. Sprinkle with salt and pepper.
Leave them undisturbed for 6 to 8 minutes, until the cut sides are deeply browned. Work in batches if needed.
Add the butter. Once melted, add the garlic and stir for about 30 seconds, just until fragrant.
Spread the sprouts in one layer on the prepared baking sheet.
Bake for 8 to 10 minutes, until a fork pierces the sprouts but they still hold their shape.
Sprinkle the mozzarella or provolone evenly over the sprouts. Broil for 4 to 5 minutes, watching closely, until melted and golden.
Season to taste and serve hot.