Preheat a 6-quart air fryer to 380°F for 3 minutes. Mix butter, garlic, chives, zest, and 1/4 teaspoon salt.
Pat thighs dry; carefully loosen skin, keeping edges attached. Thinly spread half the butter beneath each flap.
Rub outside with oil, remaining salt, and pepper; space skin-side up in basket.
Air fry without turning 25 to 30 minutes until skin golden and thickest meat 175 to 185°F, always at least 165°F. Check beside bone without touching it.
Rest uncovered, skin-side up, 5 minutes; serve one thigh per person.