First prepare the sauce by combining the mayonnaise with the dill mustard in a small bowl. Set this aside until you are ready to use it.
To prepare the eggs it's best to make one at a time in a small 6-inch skillet. If you have more than one of these small pans, you can save a little time.
First preheat pan over low heat. Add 1/4 teaspoon of butter. Add 1/2 teaspoon of minced green pepper along with 1/2 teaspoon minced white onion to the pan and saute for a couple minutes, or until softened.
Beat an 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.
Add the egg to the pan with the vegetables. Swirl the pan so that the egg spreads out.
As the egg begins to cook, use a spatula to pull in a couple of the edges so that raw egg flows around 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-omelet.
Repeat with peppers, onions and one egg EACH until you have four square mini omelets.
Flip the little omelets over and turn off the heat.
Press the sausage into four 2-ounce patties approximately the size of the bagel. Cook the sausage in a large skillet over medium heat until brown. Drain when done.
Slice a bagel in half and place it with the faces up on a baking sheet. Grill the faces of the bagel halves in your oven set on broil until golden brown. 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 American cheese onto the face of the top bagel half. Place a slice of Monterey Jack cheese over the face of the bottom bagel half. Place a sausage patty on the cheese on the bottom bagel half. Place the finished omelet onto the sausage 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.