Prepare the Sauce:

• In a small bowl, mix together the mayonnaise, mustard, and dried dill.
• Refrigerate the sauce until ready to use.

Sauté the Onions:

• Heat 1 tablespoon of butter in a large skillet over medium heat.
• Add the sliced onions and a pinch of salt.
• Stir and cook until the onions begin to soften and turn slightly brown, about 5–7 minutes.
• Transfer the cooked onions to a plate.

Scramble the Eggs:

• In the same skillet (add another tablespoon of butter if needed), pour in the beaten eggs.
• Cook over medium heat, stirring gently to scramble until just set.
• Divide the scrambled eggs evenly onto two portions for the sandwich.

Cook the Steak:

• Place the shaved steak in a bowl.
• Drizzle with Worcestershire sauce and sprinkle the Montreal steak seasoning over the top.
• Toss gently to coat the steak evenly.
• In the skillet, melt the remaining tablespoon of butter over medium heat.
• Add the seasoned steak and cook until just browned, about 2–3 minutes.
• Return the sautéed onions to the skillet with the steak and stir to combine.
• Divide the steak-onion mixture evenly into two portions and top each with a slice of American cheese, allowing the cheese to melt.

Toast the Bagel:

• Toast the bagel halves either on the skillet or in a toaster until golden and crisp.

Assemble the Sandwich:

• Spread a generous amount of the prepared sauce on both halves of the toasted bagel.
• On the bottom half, layer one portion of scrambled eggs, followed by the steak, onion, and melted cheese mixture.
• Top with the other half of the bagel.

Serve:

• Serve immediately while hot for the best flavor and texture.