Preheat the oven. Position the oven rack in the center and preheat to 400°F (200°C). Line a baking sheet with parchment paper suitable for this temperature.
Cook the bacon. Cook the bacon in a skillet over medium heat until browned and crispy. Transfer to a paper-towel-lined plate to remove excess grease.
Prepare the scrambled eggs. Scramble 5 large eggs in a nonstick skillet until just set. Remove from the heat and allow them to cool slightly before assembling the Stromboli.
Roll out the pastry. Lightly flour a clean work surface. Unfold the thawed puff pastry and gently roll it into a 14 × 14-inch square. Pinch any seams created by the folds to seal them together.
Arrange the bacon. Place 4 bacon slices in a horizontal row near the upper portion of the pastry and another 4 slices in a parallel row below, leaving approximately a 1-inch border around the edges.
Add the eggs and cheese. Spread the scrambled eggs between the bacon rows, keeping the filling away from the pastry edges. Sprinkle the shredded sharp cheddar cheese evenly over the filling.
Prepare the egg wash. In a small bowl, whisk the remaining egg with 1 teaspoon water. Lightly brush the mixture around the pastry perimeter.
Roll the Stromboli. Starting from one side, carefully roll the pastry over the filling to create a compact log. Pinch the seam closed and tuck the ends underneath to seal.
Prepare for baking. Transfer the Stromboli to the prepared baking sheet, seam-side down. Brush the entire surface with the remaining egg wash and lightly season with salt and pepper.
Bake until golden. Place in the preheated oven and bake for 25–30 minutes, or until the puff pastry is deeply golden, crisp, and cooked through. The egg filling should reach 165°F (74°C).
Rest before slicing. Remove from the oven and allow the Stromboli to rest for at least 5 minutes.
Slice and serve. Cut into six portions using a sharp serrated knife and serve warm.