Step 1

Heat the oil in a medium nonstick pan over medium heat until shimmering. Add the sausage and onion and cook, breaking up the meat with a wooden spoon into small pieces and stirring occasionally, until the sausage is cooked through, about 8 minutes. Meanwhile, place the 4 eggs and the salt in a medium bowl, season with pepper, and whisk to combine.

Step 2

Reduce the heat to medium-low. Pour in the egg mixture. Using a rubber spatula, scramble the eggs, stirring occasionally and pushing the cooked parts from the edges of the pan to the center, until they are almost set but not dry, about 3 minutes. Transfer to a large plate and spread into an even layer to cool while the oven is heating (the eggs will continue to set).

Step 3

Arrange a rack in the lower third of the oven and heat to 425°F. Make the egg wash by beating the remaining egg and water in a small bowl together; set aside. Line a baking sheet with parchment paper; place the parchment on a work surface.

Step 4

When the oven is ready, roll the pizza dough out into a rough 12x14-inch rectangle. Transfer it onto the parchment. Position the parchment with the dough on it so a short end is facing you. Sprinkle 1/2 of the cheese lengthwise down the center of the dough, creating a strip of cheese that is 12 inches long and 3 1/2 inches wide, leaving a 1-inch border at each short end and a 4 1/4-inch border on each long end. Spoon the egg mixture evenly over the cheese. Sprinkle with the remaining cheese and parsley.

Step 5

Using a very sharp small knife or pizza cutter, cut 1 1/2-inch-wide strips on either side of the filling (as if you were making a thick fringe on either side of it), stopping the cut about 1/4-inch from the filling.

Step 6

Starting with the bottom left strip, fold it up and over the filling at a slight angle. Next, fold the bottom right strip up and over the filling, meeting and slightly crossing the first strip, making a sort of crisscross or chevron pattern. Continue alternating left and right strips so you wind up with a crisscross that covers most of the filling. Press the dough strips together firmly without squishing the filling so the strips stay together when baked.

Step 7

Seal the stromboli by pinching together the uncovered ends and the strips closest to them. Grasping the parchment, transfer the stromboli and parchment onto the baking sheet. Liberally brush a layer of the egg wash on the stromboli.

Step 8

Bake until the stromboli is golden-brown, about 20 minutes. Let the stromboli cool for 5 minutes before cutting crosswise and serving.