Preheat either the air fryer to 400°F or the oven to 450°F, depending on which method you will use. If using the oven, line a baking sheet with foil or parchment paper and set aside.
Cut each cleaned and dried baby potato in half or quarter larger potatoes so all pieces are roughly similar in size.
In a large bowl, combine 2 tablespoons oil of choice, 2 tablespoons melted butter, 1/4 cup grated Parmesan cheese, and 1/2 teaspoon each Italian seasoning, garlic powder, salt, and black pepper. Whisk until blended.
Add the cut potatoes to the bowl and toss well so all pieces are evenly coated with the oil/butter/cheese/seasoning mixture.
Air fryer method: Place the coated potatoes in a single layer in the air fryer basket (do not overcrowd; cook in batches if needed). Cook at 400°F for 15 minutes, shaking or stirring the basket about halfway through (around 7–8 minutes). Cook until the potatoes are golden and crispy outside and tender inside.
Oven method: Spread the coated potatoes in a single layer on the prepared baking sheet. Bake at 450°F for about 20 minutes, stirring or turning the potatoes once about halfway through (around 10 minutes), until the outsides are golden and crispy and the insides are tender.
Remove the potatoes from the air fryer or oven and serve immediately. If desired, sprinkle additional grated Parmesan before serving.