Prepare the breakfast sauce by mixing together mayonnaise, mustard, and dill in a small bowl. Store in the refrigerator until ready to use.
Heat a skillet over medium heat and melt 1 tablespoon of butter. Add the sliced onions and a pinch of salt. Cook until the onions are slightly browned, then remove and set aside.
In a bowl, mix the shaved steak with Worcestershire sauce and Montreal steak seasoning. Heat another tablespoon of butter in the skillet and cook the steak until just browned. Return the onions to the skillet, mix with the steak, and form two piles. Place a slice of cheese on each pile to melt.
In the same skillet, add the last tablespoon of butter and scramble the beaten eggs, dividing them into two portions for the sandwiches.
Toast the bagels on a skillet or in a toaster. To assemble the sandwiches, spread some of the prepared sauce on each bagel half. Place half of the scrambled eggs on the bottom half of each bagel, then top with the steak and onion mixture, and cover with the top half of the bagel.