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