Preheat the oven to 375°F (190°C) and line a baking tray with parchment paper.
Whisk the self-rising flour and vanilla protein powder together in a large bowl.
Add the Greek yogurt and mix until a soft, slightly tacky dough forms.
Fold 2 tablespoons of rainbow sprinkles gently into the dough.
Divide the dough into 4 equal portions.
Shape each portion into a thick disk and make a 1½–2-inch hole through the center.
Arrange the bagels on the prepared baking tray and brush each one lightly with beaten egg.
Sprinkle the remaining 1 tablespoon of rainbow sprinkles over the tops.
Bake for 20–25 minutes, until the bagels are set and lightly golden.
Cool on the tray for 5 minutes, then transfer to a wire rack for another 10–15 minutes before slicing.