Preheat oven to 425°F. Melt the butter and set aside. Line a baking sheet with parchment paper.
Wash the potatoes well and slice them in half.
Place the halved potatoes in a bowl. Add the garlic powder, oregano, salt, and pepper, then drizzle with the melted butter. Toss until well coated, making sure the cut sides are buttered.
Place the parmesan in a shallow bowl. Press the cut side of each potato into the parmesan, then arrange cut-side-down on the prepared baking sheet. (Or sprinkle small bundles of parmesan onto the pan and set a potato cut-side-down on each.)
Bake at 425°F for 25–30 minutes, until fork tender and the bottoms are golden and crispy.
Remove the potatoes from the pan right away while warm. Serve with the chilled herb dip.