Combine mayonnaise with the dill mustard in a small bowl. Set this aside until you are ready to use it.
Heat 1 teaspoon of vegetable oil in a medium skillet over medium heat. Separate the onion slices and saute in the oil until light brown. Remove and set aside.
Break up the sandwich steak into the hot pan and cook until brown. Drain off fat. Add the grilled onions to the meat and stir.
(Tip: To make the eggs it's best to make one at a time in a small 6 inch skillet.) Beat one egg in a small bowl with a whisk until it is smooth, but not foamy. Add a pinch of salt and pepper to the egg.
Heat a small 6 inch skillet over low heat. Add 1/4 teaspoon of butter to the pan. When the butter has melted add the egg to the pan. Swirl the pan so that the egg spreads evenly.
As the egg begins to cook, use a spatula to pull in a couple of the edges so that raw egg flows from the top onto the hot pan. Cook for 2 to 3 minutes, then fold over one of the edges of the egg using a spoon or fork. Fold it down about an inch. Fold the opposite end over as well. Then fold the remaining two edges over, creating a small rectangular or square mini-omelette. Flip the little omelette over and turn off the heat.
Slice bagels in half and place faces up on a baking sheet. Grill the faces of the bagel halves in your oven set on broil until golden brown. (Tip: You may also use a toaster oven for this step, but be sure to place the sliced bagel halves onto a small baking sheet or on aluminum foil.).
When the bagels are toasted, spread about 1/2 tablespoon of the sauce onto the face of the top bagel half. Place a slice of cheese onto the face of each bagel half.
Divide the meat into four portions and stack one portion onto the cheese on the bottom bagel half.
Place the finished omelette onto the meat on the bottom half of the sandwich. Finish by flipping the top half of the sandwich over onto the bottom. Heat for 15 seconds in microwave if needed to warm.