Melt the cheeses. Combine mozzarella and cream cheese in a large microwave-safe bowl. Microwave in 30-second bursts, stirring between each, until fully melted and stretchy (about 90 seconds total).
Mix the dough. Working quickly while the cheese is hot, add almond flour, baking powder, vinegar, and egg. Stir with a silicone spatula until a uniform sticky dough forms.
Rest and divide. Let the dough cool 3–4 minutes, then divide into 6 equal portions (about 2.5 oz each).
Shape. Roll each portion into a 6-inch rope, pinch the ends together to form a ring, and smooth the seam with wet fingertips.
Preheat. Preheat air fryer to 350°F (175°C). Line basket with perforated parchment or spray lightly with oil.
Brush and season. Brush bagels with melted butter and sprinkle with everything seasoning, sesame seeds, or coarse salt while butter is still wet.
First fry. Air fry at 350°F for 5 minutes.
Flip. Flip bagels with tongs, brush other side with remaining butter.
Finish frying. Air fry another 5–7 minutes until deeply golden brown and firm to the touch (internal temp 200°F).
Cool and serve. Transfer to a wire rack and cool at least 5 minutes before slicing. Serve with cream cheese, butter, or smoked salmon.