Cut zucchini into two or three 3-inch pieces, then spiralize each piece.
Whip egg in a shallow bowl until foamy. Carefully pour melted butter into egg and whip until blended.
Combine flour, oregano, garlic powder, and pepper in a second shallow bowl. Stir together panko bread crumbs and Parmesan cheese in a third bowl.
Dredge zucchini spirals in seasoned flour mixture, shaking off excess flour. Dip spirals in egg wash, allowing excess wash to drip off, and then dredge spirals in Parmesan-bread crumbs. Place on a clean sheet pan.
Set air fryer temperature to 400 degrees F (200 degrees C). Set timer for 12 minutes. Transfer breaded zucchini fries into the air fryer basket in batches without crowding. Cook until crispy, about 12 minutes. Serve immediately.