Preheat the air fryer to 375°F (190°C) for 2 minutes. Lay the bacon slices flat in a single layer in the air fryer basket.
Air fry the bacon at 375°F for 6 to 8 minutes, flipping once halfway through, until crispy. Remove and set aside on a paper towel-lined plate to drain. Pour off any excess grease from the basket.
Spread a thin layer of softened butter on one side of each bread slice. Using the back of a spoon or the bottom of a small glass, press a wide, shallow well into the center of the unbuttered side of each slice - deep enough to cradle an egg.
Thick-cut bread works best here because it creates a sturdy nest that holds the egg in place.
Place the bread slices buttered-side down in the air fryer basket. Carefully crack one egg into the well of each slice. Season the eggs with a pinch of salt and black pepper.
Air fry at 330°F (165°C) for 5 to 6 minutes, until the egg whites are just set but the yolks are still soft and runny. Keep a close eye starting at the 5-minute mark - every air fryer runs a little differently, and you want that perfect jammy yolk.
Break or cut each bacon slice in half and lay the pieces on top of the eggs. Sprinkle each toast evenly with shredded cheddar cheese. Air fry for 1 more minute at 330°F until the cheese is melted and bubbly.
Carefully lift the toasts out of the air fryer with a spatula. Sprinkle with freshly chopped chives and serve immediately - the runny yolk, salty bacon, and melted cheese are best enjoyed right away.