Place one potato between two wooden chopsticks on a board. Cut crosswise slices about 1/8 inch apart, stopping about 1/4 inch above the bottom so the base stays joined; take extra care at the narrow ends. Repeat with the second potato.
Preheat a 5-quart or larger air fryer to 350°F for 3 minutes. Rub potatoes with oil and season with salt and pepper, working a little seasoning into the cuts without forcing them apart.
Place cut-side up in the basket and air fry for 25 minutes. Transfer briefly to a plate and gently brush melted butter across the tops and between the cuts as they open.
Return to the basket, raise the temperature to 400°F and cook for 15 to 20 minutes until the slice edges are golden and a skewer passes through the joined base easily. Add 5 minutes if the base remains firm.
Rest for 3 minutes and serve one whole fanned potato per person, keeping the bottom connected.