In a medium bowl, combine self-rising flour and greek yogurt. Stir until a shaggy dough forms, then turn it out onto a floured surface and knead gently for about 1 minute until smooth. Do not overwork the dough.
Divide the dough into 4 equal pieces. Roll each piece into a rope about 8 inches long, then shape into a circle and pinch the ends together firmly to form a bagel shape.
Brush the top of each bagel with beaten egg, then press generously into everything bagel seasoning to coat the tops.
Preheat the air fryer to 330°F (165°C). Place the bagels in the basket with space between them.
Air fry for 12 to 14 minutes, until the bagels are golden brown and sound hollow when tapped on the bottom.
Brush with melted butter while still warm and serve with your favorite spread.