Preheat the oven to 425°F (220°C). Line a baking sheet with parchment paper for easy cleanup.
Wash and dry the potatoes, then cut them into evenly sized wedges for uniform roasting.
Toss the potato wedges with olive oil, salt, and black pepper until evenly coated. Arrange them in a single layer on the prepared baking sheet.
Roast the potatoes in the oven for about 20 minutes, flipping once halfway through, until they are golden brown and crispy around the edges.
Remove the baking sheet from the oven and increase the oven temperature to broil. Drizzle melted butter over the hot potatoes for extra crispness and flavor.
Sprinkle the shredded cheese evenly over the hot, crispy potatoes. Return the sheet to the oven, broiling until the cheese is melted, bubbly, and slightly browned, about 2-4 minutes.
Remove the cheesy potatoes from the oven and let sit for a minute to settle. Serve immediately for the best gooey, crispy contrast.