Leavened dough:
Pour the water into a bowl and add the yeast and sugar. Stir until dissolved.
Let it rest until the mixture becomes frothy (about 10 minutes).

In the mixing bowl, fitted with a hook attachment, combine the flour, salt, egg, and butter.
Mix gently, then add the yeast mixture.
Knead for about 8 minutes, or until the dough is smooth and non-sticky.
Cover the bowl with plastic wrap and let it rise for 1 hour, or until it has slightly risen.

Filling:
Melt 2 tablespoons of butter in a large skillet over medium heat.
Add the leeks and cook until they are tender, about 3 minutes.
Remove the leeks from the skillet and set aside.
Add the remaining butter to the skillet, then add the pleurotes, white mushrooms, brown mushrooms, and garlic.
Cook until the mushrooms are tender.
Return the leeks to the skillet, then add the heavy cream and soy sauce.
Bring to a boil, then cook for 30 seconds.
Remove from heat, add the cheese and parsley. Season to taste. Let it cool.

Preheat the oven to 400 F (200 C).
Roll out the dough into a 14-inch (35 cm) circle on a floured surface.
Place the dough on a baking sheet lined with parchment paper.
Spread the leek and mushroom mixture evenly over the dough, leaving a 1-inch (2.5 cm) border.
Fold the dough over the filling, forming a border.

Mix the egg yolk and heavy cream, then brush the dough with the mixture.
Bake for 20-25 minutes, or until the dough is golden brown and the filling is hot.