Combine yogurt, mustard, garlic powder, and cayenne in a small bowl. Cover and refrigerate mustard dipping sauce until ready to use.
Preheat an air fryer to 400 degrees F (200 degrees C).
Place flour on a flat plate. Divide sausage into 12 equal balls. Using your hands, flatten each ball into a paper thin patty. Dust both sides of each patty with flour; the flour on one side will help the sausage adhere to the egg and the flour on the other side will keep the sausage from sticking to your hands.
Shape each sausage patty around a hard-boiled quail egg, making sure the egg is completely sealed inside. Dip each in the beaten egg and coat in bread crumbs.
Spray the air fryer basket with nonstick cooking spray. Place 6 eggs in the basket and mist the tops with cooking spray. Cook for 8 minutes. Flip and spray again. Cook 6 additional minutes. Repeat with remaining eggs.
To serve, slice each egg in half and serve with mustard dipping sauce.
