In a mixing bowl, combine the self-rising flour and Greek yogurt. Stir until a shaggy dough forms.
Transfer the dough to a lightly floured surface and knead for 1 to 2 minutes until smooth. If the dough is sticky, add a small amount of flour as needed.
Divide the dough into 4 equal pieces. Roll each piece into a rope about 6 to 7 inches long, then join the ends and pinch to seal, forming a bagel shape.
Brush the tops of the bagels with the beaten egg. This helps create a golden, bakery-style finish.
Lightly grease the air fryer basket or line it with parchment paper. Arrange the bagels in a single layer, leaving space between them.
Air fry at 330°F (165°C) for 10 to 12 minutes, or until the bagels are puffed, golden brown, and cooked through.
Remove the bagels from the basket and let them cool for 5 minutes before slicing and serving.