In a small bowl, mix together the mayonnaise, mustard, and dried dill until well combined. Cover and store in the refrigerator until ready to use.

Melt 1 tablespoon of butter in a skillet over medium heat. Add the sliced onions, sprinkle with salt, and cook, stirring occasionally, until they turn slightly brown and fragrant. Remove from the skillet and set aside.

In the same skillet, add another tablespoon of butter. Once melted, pour in the beaten eggs and scramble until just set. Divide the eggs into two portions for your sandwiches.

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

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

Spread the sauce on each half of the toasted bagels. Add one portion of scrambled eggs to the bottom half, then layer with the steak, onion, and cheese mixture. Place the top half of the bagel on each to complete the sandwich.