Preheat the oven to 425°. Place the zucchini in a bowl and season with salt; let stand for 5 minutes. Squeeze with paper towels to dry. In a medium bowl, beat together 5 eggs, the oregano and 1/2 teaspoon each salt and pepper. In a small bowl, beat together the remaining egg with 1 tablespoon water.
In a large nonstick skillet, melt the butter over medium heat. Pour in the oregano-egg mixture and scramble for 1 minute; transfer to a plate.
On a work surface, brush the edges of the puff pastry with some egg wash. Working with 1 piece at a time, arrange one-quarter of the scrambled eggs on one side of the pastry and top with one-quarter each of the zucchini and cheese, then 1 tomato slice. Fold over the pastry and press the edges to seal; transfer to a parchment-lined baking sheet and brush with more egg wash. Repeat with the remaining ingredients to make 3 more pastries.
Bake until the pastries are puffed and golden-brown, 20 to 25 minutes.