Preheat your oven to 400°F. While it's warming up, drizzle olive oil over a 9x13-inch baking dish, tilting the dish to coat the bottom evenly.
Combine the grated Parmesan, garlic powder, oregano or thyme, paprika, salt, and black pepper in a bowl.
Sprinkle the Parmesan mixture evenly into the oiled baking dish, covering the bottom to form an even layer. Don't touch it after sprinkling.
Place the halved potatoes cut-side down onto the Parmesan bed, pressing them down slightly to ensure they're well-coated. Optional: Spray or brush the tops of the potatoes with a bit more olive oil.
Roast for about 35-40 minutes. The potatoes are done when they are fork-tender and the Parmesan is crispy.
Once out of the oven, let the potatoes rest in the dish for about 5 minutes. Garnish with chopped green onions and serve with sour cream for dipping.