Preheat oven to 375°F. Line a baking sheet with parchment paper.
Heat a medium sized non-stick skillet over medium heat. Add the beaten eggs and cook over medium low heat, stirring with a rubber spatula, until cooked to your liking. Remove from heat and set aside
Using a rolling pin, roll the pizza dough into a rectangle approximately 18 inches by 12 inches. Use a pizza cutter to roll the dough into 6 pieces. You'll have six equal sized squares.
Place a sausage patty on the center of each dough square. Top the sausage patties with equal portions of scrambled eggs.
Sprinkle shredded cheddar cheese over the center of each.
Stretch the dough for each piece and bring the corners and sides up to the top of each bagel bomb. You can use your fingers to pinch the seams shut. You may need to pull and manipulate the dough to get around the toppings.
Use a silicone pastry brush to brush the remaining beaten egg over the top.
Bake in the oven for about 30 minutes, until tops are light golden brown. You may want to rotate your pan halfway through, as ovens have different hot spots.
Cool on the pan for 5-10 minutes before removing the bagel bombs to a wire cooling rack.