In a small bowl, combine the mayonnaise, mustard, lemon juice, paprika, and dried dill. Mix well to create a flavorful sauce. Set aside.
In a medium-sized bowl, season the shaved ribeye steak with Montreal steak seasoning, salt and pepper. Toss to coat the steak evenly.
Heat the vegetable oil in a large skillet over medium-high heat. Add the seasoned steak and sliced onions. Cook, stirring occasionally, until the steak is browned and cooked through, and the onions are soft and caramelized. This should take about 5-7 minutes. Remove the steak and onions from the skillet and set aside.
In the same skillet, melt the butter over medium heat. Crack the eggs into the skillet, season with salt and pepper, and fry them until the whites are set and the yolks are still runny. Remove from heat and set aside.
To assemble the sandwich, spread a generous amount of the sauce onto the bottom half of each toasted bagel. Layer the cooked steak and onions onto the sauce. Place 2 slices of American cheese on top of the steak and onions, allowing the residual heat to melt the cheese slightly. Add a fried egg on top of the cheese. Finally, place the top of the bagel on the stack to complete the sandwich.
Repeat the process for the remaining 3 bagels. Serve immediately and Enjoy!