Preheat the oven to 425°F.

Cut the potatoes in half lengthwise. Using a sharp knife, lightly score the cut side of each potato in a shallow crosshatch pattern, being careful not to cut deeply enough for the potatoes to fall apart.

Arrange the potatoes cut-side up on a microwave-safe plate. Microwave on HIGH for 5–7 minutes, until the potatoes are beginning to soften but are still firm enough to handle. Very large potatoes may need another 1–2 minutes.

Pour the melted butter and 1 tablespoon olive oil into a 9-by-13-inch metal baking pan and tilt to coat the bottom. Sprinkle the Parmesan evenly over the butter mixture. Sprinkle the garlic powder, onion powder, smoked paprika, 3/4 teaspoon kosher salt, and black pepper evenly over the Parmesan. Do not stir.

Carefully place the warm potato halves cut-side down onto the Parmesan mixture, pressing them gently into the cheese. Brush the skins with the remaining 1 tablespoon olive oil and sprinkle with the remaining 1/4 teaspoon kosher salt.

Bake undisturbed for 20–25 minutes, until the potatoes are completely tender when pierced with a thin knife and the Parmesan underneath has formed a deep golden-brown crust.

Remove the pan from the oven and let the potatoes rest undisturbed for 5 minutes so the Parmesan crust firms up and adheres to the potatoes.

Slide a thin metal spatula firmly underneath each potato, making sure to capture the Parmesan crust. Turn the potatoes cut-side up and serve hot, either as-is or with optional toppings.