Pour water into a large saucepan and bring to a boil over high heat. Drop noodles in and boil for 3 minutes. Remove from heat and let sit for 2 minutes. Drain.
Preheat an air fryer to 400 degrees F (200 degrees C).
Meanwhile, whisk egg and 1/2 of the reserved seasoning packet together in a bowl. Gently stir in noodles until they are evenly coated. Divide noodles into 2 small round containers, which helps make more uniform buns.
Spray the basket of the air fryer with nonstick spray. Dump one portion of noodles into the basket. Flatten the noodle mixture with a spatula to form the bun. Cook for 4 minutes. Flip the bun over using a spatula and cook until desired brownness, about 4 minutes more. Transfer to a plate and repeat with remaining noodles.
Sprinkle remaining seasoning over the beef and form into a patty. Air fry for 4 minutes. Flip and cook for 4 minutes more. Sandwich the patty between the ramen buns.