Preheat oven to 200°C (400°F) and line a baking sheet with parchment paper.
Heat olive oil in a large skillet over medium-high heat; add ground beef and cook until browned, breaking into small crumbles.
Add diced onion, mushrooms, and garlic to the skillet; sauté for 6-8 minutes until moisture from mushrooms has evaporated.
Stir in grated zucchini, salt, pepper, and oregano; cook for 2 minutes and remove from heat to cool slightly.
Roll the puff pastry sheet on the parchment paper into a 10x12 inch rectangle.
Place the beef and vegetable mixture in a vertical column down the center third of the pastry.
Top the filling evenly with shredded mozzarella cheese.
Cut 1-inch wide diagonal strips into the remaining side thirds of the pastry, from the filling to the outer edge.
Fold the strips over the filling, alternating sides to create a braided appearance, and tuck the ends under.
Brush the surface of the braid thoroughly with the beaten egg wash.
Bake for 25 to 30 minutes until the pastry is deeply golden and puffed.
Allow the braid to rest for 5 minutes before slicing into sections for service.