Set a potato between two wooden chopsticks. Slice crosswise 1/8 inch apart, stopping 1/4 inch above bottom to keep base joined; take care at narrow ends. Repeat with second potato.

Preheat a 5-quart or larger air fryer to 350°F for 3 minutes. Rub potatoes with oil; add salt and pepper, working some into cuts without forcing apart.

Air fry cut-side up 25 minutes. Briefly transfer to a plate; gently brush melted butter over tops and into opening cuts.

Return to basket at 400°F; cook 15 to 20 minutes until edges golden and joined base easily pierced. Add 5 minutes if base remains firm.

Rest 3 minutes. Serve one whole fanned potato per person, bottom connected.