In a small bowl, mix together mayonnaise, mustard, and dried dill. Store the mixture in the refrigerator until ready to use.

Melt 1 tablespoon of butter in a skillet over medium heat. Add sliced onions and a pinch of salt, cooking until they turn slightly brown. Remove the onions and set aside.

Add another tablespoon of butter to the same skillet. Pour in beaten eggs and scramble them until cooked through. Divide the eggs into two portions for each bagel.

In a bowl, toss shaved steak with Worcestershire sauce and Montreal steak seasoning. Melt the last tablespoon of butter, add the steak, and cook until browned. Return the onions to the skillet and mix with the steak. Separate into two piles and place one slice of cheese on each pile to melt.

Toast bagels on the skillet or in a toaster until golden brown.

Spread the sauce on each half of the bagels. Add scrambled eggs to the bottom half, then top with the steak, onions, and melted cheese. Place the top bagel half on each to complete the sandwich.