Finely chop the steak. Mix with Worcestershire, salt, pepper, garlic powder and onion powder. Divide into 4 portions and press each into a thin patty, about 4 inches wide and 1/4 inch thick.
Preheat the air fryer to 400°F. Air fry the patties in a single layer for 5 to 6 minutes, flipping halfway, until they reach 160°F inside.
Meanwhile, melt half the butter in a nonstick skillet over medium heat. Cook the onion with a pinch of salt for 5 to 6 minutes, until soft and golden. Transfer to a plate.
Whisk the eggs, milk, salt and pepper. Add the remaining butter to the skillet and lower heat to medium-low. Pour in half the egg mixture and cook without stirring until set, about 2 minutes. Fold the sides in to form a square, cut in half, and transfer to a plate. Repeat with the remaining eggs.
Spread softened butter on the cut sides of the bagels. Air fry cut side up at 360°F for 2 to 3 minutes, until golden.
Spread breakfast sauce on both bagel halves. Layer the steak patty, onions, cheese and folded egg on each bottom half. Close and serve hot.