Preheat the oven to 400°F (205°C). Fill a saucepan halfway with water, add the potatoes, and boil for 5 minutes. Drain the potatoes and transfer them to a cutting board. Cut the potatoes in half and set aside.
Combine melted butter, half of the grated cheese, salt, ground black pepper, Italian seasoning, and paprika in a bowl. Mix well. Pour the mixture into a 9" x 9" (23 x 23 cm) square baking tray and evenly distribute it using a brush.
Transfer the potatoes to the baking tray in a single layer, with the cut sides facing down. Sprinkle the remaining Parmesan cheese on top of the potatoes. Bake for 15 to 20 minutes, or until golden brown and tender inside.
Serve immediately, topped with additional grated Parmesan cheese if desired.