Preheat the oven to 350 degrees Fahrenheit. Line a baking sheet with a silicone baking mat or parchment paper, or lightly coat it with non-stick cooking spray.
In a large mixing bowl, combine the self-rising flour and Greek yogurt. Stir with a spoon until the mixture begins to come together, then use your hands to mix until a dough forms.
Transfer the dough to a lightly floured surface. Knead for 2 to 3 minutes until smooth but still slightly tacky.
Divide the dough into 4 equal portions.
Roll each portion into a rope about 10 inches long. Form each rope into a circle, pinching the ends together firmly to create a bagel shape.
Arrange the bagels on the prepared baking sheet.
In a small bowl, whisk together the egg and water to make an egg wash. Brush the tops and sides of each bagel with the egg wash.
Sprinkle with everything bagel seasoning, sesame seeds, poppy seeds, coarse salt, or leave them plain.
Bake for 25 minutes, or until the bagels are puffed and golden brown.
Remove from the oven and cool on a wire rack for 10 minutes before slicing.
Serve warm or toasted with cream cheese, butter, jam, or your favorite spread.