01 -
Stir together the mayonnaise, yellow mustard, steak sauce, lemon juice, dill, and black pepper. Refrigerate while you cook the sandwich fillings.
02 -
Toss the thin steak with Worcestershire sauce, garlic powder, 1/2 teaspoon salt, and black pepper. Let it sit while the onions start cooking.
03 -
Melt 1 tablespoon butter in a large skillet over medium heat. Add the sliced onion and cook for 7-9 minutes, stirring often, until soft and golden. Transfer to a plate.
04 -
Increase the heat to medium-high. Add 1 tablespoon butter and the steak in an even layer. Cook for 2-4 minutes, turning once or twice, until browned and cooked through. Move the steak to the plate with the onions.
05 -
Beat the eggs with the remaining 1/4 teaspoon salt. Melt the remaining 1 tablespoon butter in a nonstick skillet over medium-low heat, add the eggs, and cook until just set. Fold into four squares.
06 -
Toast the split bagels until the cut sides are golden. Spread sauce on both sides of each bagel.
07 -
Layer one slice of cheese, steak, onions, folded egg, and a second slice of cheese on each bottom bagel. Add the tops and let the hot fillings melt the cheese for 1 minute before serving.