In a mixing bowl, whisk together the all-purpose flour, whole wheat flour (or extra all-purpose), baking powder, and kosher salt until evenly combined.
Make a well in the center of the dry ingredients and add the Greek yogurt. Stir with a spoon or rubber spatula until a shaggy dough forms.
Turn the dough onto a lightly floured surface and knead 1 to 2 minutes until smooth; dust hands and counter with a little flour only if the dough is too sticky.
Divide the dough into 4 equal pieces. Roll each into a ball, then roll into an 8–9 inch rope about 3/4 inch thick. Join the ends and press gently to seal to form a bagel.
Place bagels on a parchment-lined baking sheet if adding toppings. Brush the tops with the beaten egg mixed with 1 tablespoon water, then sprinkle with desired toppings (everything bagel seasoning, cinnamon sugar, rosemary Parmesan, or shredded cheese).
Arrange the bagels in a single layer in the air fryer basket without touching (cook in batches if needed). Air fry at 350°F for 8 to 10 minutes, until golden brown on top and the sides.
Transfer the bagels to a wire rack to cool slightly, then serve warm.