Separate the tops and bottoms of the bagels. Spread about 1/2 tablespoon of butter on the cut side of each bagel half and place them butter-side down in a skillet over medium heat. Cook for 1 to 2 minutes, until lightly toasted. Remove the bagels from the skillet and set aside.
Add 1 tablespoon of butter and the onions to the skillet. Sauté for 2 to 3 minutes, until the onions become translucent. Remove the onions from the skillet and set aside.
In a mixing bowl, combine the ground chuck, Worcestershire sauce, salt, garlic powder, and ground black pepper. Mix with your hands until well blended. Form the mixture into one large ball. Divide the ball in half and then in half again to create 4 equal sections. Form each section into a flat, round patty about 1/2 inch thick (they will expand slightly while cooking).
Place the skillet over medium-high heat and cook the patties for about 3 minutes on each side, until fully cooked through. Remove the patties from the skillet. Discard the grease and wipe the skillet with a paper towel.
Reduce the heat slightly and spray the skillet with nonstick spray. Pour the whisked eggs into the skillet and allow them to cook undisturbed for about 2 to 3 minutes. When the eggs are cooked on the bottom, use a spatula to fold the eggs over, creating a half circle. Cook for another 1 to 2 minutes, until the eggs are cooked through.
Place 2 slices of cheese side by side over the eggs and cook until the cheese starts to melt slightly. Cut the eggs in half, creating 2 quarter circles. Remove from the heat and set aside.
In a small mixing bowl, combine the mayonnaise, mustard, and dill. Mix well.
To assemble the sandwiches, spread the mayonnaise mixture on all 8 faces of the toasted bagels. Add a steak patty, onions, egg, and cheese to the bottom halves of the bagels and cover with the top halves.
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Recipe Tips
For a fluffier look and taste, add a few tablespoons of water to the eggs when you beat them.