Prep the pumpkin. Place the pumpkin purée in a fine mesh strainer for 5 minutes to drain excess liquid.
Mix the dry ingredients. In a large bowl, whisk together the self-rising flour, pumpkin pie spice, brown sugar, and salt.
Add the wet ingredients. Stir in the Greek yogurt and drained pumpkin purée until a shaggy dough forms.
Knead. Turn out onto a lightly floured surface and knead 6 to 8 times until smooth. Add flour 1 tablespoon at a time if sticky.
Shape. Divide into 4 equal pieces. Roll each into an 8-inch rope, then bring the ends together and pinch firmly to form a bagel ring.
Egg wash. Brush each bagel all over with the egg wash. Sprinkle generously with pumpkin spice sugar if using.
Air fry. Preheat air fryer to 325°F. Lightly spray the basket, arrange bagels in a single layer with space between them, and air fry for 14 to 16 minutes, until deeply golden brown.
Cool. Transfer to a wire rack and let cool for at least 10 minutes before slicing. Serve with cream cheese, butter, or pumpkin cream cheese.