Combine mayonnaise, mustard, and dried dill in a small bowl and refrigerate until needed.
Melt 1 tablespoon butter in a skillet over medium heat, add sliced onions sprinkled with salt, and cook until lightly browned. Remove and set aside.
Add another tablespoon butter to the skillet, pour in beaten eggs, scramble, and divide into two portions.
In a bowl, toss shaved steak with Worcestershire sauce and Montreal steak seasoning.
Melt the last tablespoon butter in the skillet, add steak and cook until browned.
Return cooked onions to the skillet, mix with steak, and divide into two piles. Top each with a slice of cheese to melt.
Toast bagel halves until golden brown.
Spread the prepared sauce on each half of the bagels.
Assemble sandwiches by adding scrambled eggs to the bottom half of each bagel, followed by the steak, onion, and cheese mixture, then top with the other half of the bagel.