For the sausage: Puree the onion in a food processor and transfer to a bowl. Add the garlic, mustard, paprika, oregano, salt and pepper. Stir until blended. Put the sausage in a large mixing bowl and break up into small pieces. Mix in the onion mixture, using your hands to blend into the sausage. The puree should be blended evenly with the sausage and you should see no clumps. Form the sausage into 6 balls, a bit bigger than a golf ball, and place on a dinner plate or small baking pan. Be sure to roll and form the balls tightly.
For cooking the sandwiches and assembling: Preheat a cast-iron griddle or large pan over medium heat. Be sure your griddle is fully heated before beginning. The egg and sausage should sizzle when they hit the griddle. Spread some butter down the center of each bread slice using a brush, and then place butter-side down on the griddle. Evenly spread a heaping tablespoon of the pesto onto the bread using a spatula, covering the bread right up to the edges. Sprinkle as much Parmesan as you desire on top of the pesto. At the cart, we typically use about an 1/8 cup on each sandwich.
Place the sausage balls on the griddle and press into a patty using a strong metal spatula. You will be covering one-half of the bread with this patty, so match the patty size to the bread.
Spread a thin layer of butter onto the griddle, and then crack your eggs onto that spot. Sprinkle a small pinch of black pepper and cayenne on each egg. Check your sausage patty; when one side is charred, flip. When the whites of your egg start turning opaque, flip your egg. Let the sausage and egg cook for about 2 minutes after flipping. Then place your sausage on one half of the bread and top it with the egg. Use a knife to cut the bread in half and fold it over to create a sandwich. Your bread should have a nice brown toasted mark down the center.
Use your judgment on how many sandwiches you can cook at once. You can also experiment with toasting/topping all the bread first, then cooking all the sausages, followed by all the eggs, etc. Or you can cook them 1 or 2 at a time. It's up to you!