Using a round cookie cutter or the rim of a glass, press a circle about 3 inches wide into the center of each bread slice to create a well. Push the bread down in the center to form a shallow cup, but do not cut all the way through. You want a nest that will hold the egg.
Spread the softened butter evenly on the top surface and edges of each bread slice. This helps the toast crisp up beautifully in the air fryer.
Preheat the air fryer to 340 degrees F for 2 minutes.
Place the bread slices in the air fryer basket. Carefully crack one egg into the center well of each slice. Season the eggs with salt, black pepper, and a pinch of smoked paprika.
Cook at 340 degrees F for 5 to 6 minutes. At this point the whites should be mostly set. Sprinkle the shredded cheddar cheese around the edges of each toast.
Cook for another 2 to 3 minutes until the cheese is melted and bubbly and the egg whites are fully set. For a runny yolk, pull them at 2 minutes. For a firmer yolk, go the full 3 minutes.
Carefully remove the egg toasts from the air fryer, top with fresh chives, and serve immediately.