Preheat the oven to 400°F (200°C). Line a baking sheet with parchment paper.

On a lightly floured surface, unroll one puff pastry sheet. Use a 3-1/4-inch round cutter to cut out 9 circles. Space them out on the baking sheet. Reroll the scraps of the puff pastry and cut out 9 more. You will also need to use the second sheet of puff pastry.

Then, with a 2-1/4-inch round cutter, cut the centers out of the second set of circles. Place those rings on top of the circles that are on the baking sheet. You can also cook the smaller center circles on the baking sheet.

In a small bowl, beat together the egg and water to make an egg wash. Brush the egg wash over each piece of pastry, then chill them for 15 minutes in the refrigerator. Bake for about 20 minutes, until they’re golden brown. Remove from the oven and cool on a wire rack.

While the pastries are cooking, you can make the filling. Heat olive oil in a skillet over medium heat. Once hot, add the mushrooms, oregano, salt and pepper. Saute, stirring occasionally, until mushrooms are golden. Transfer to a plate.