Sift the all-purpose flour into a large bowl. Add the dry yeast, salt, and sugar. Mix well to combine.
Pour in the lukewarm water and olive oil, then start kneading. Knead until you have a smooth, elastic dough. Cover and let rise in a warm place for 1 hour.
While the dough is rising, prepare the filling. Slice the mushrooms and finely chop the garlic.
Heat a little olive oil in a skillet, add the garlic and mushrooms. Sauté for 5-6 minutes, until the mushrooms release their liquid and turn golden brown. Season with oregano, basil, and black pepper.
Divide the risen dough into two equal portions and roll each out into a circle on a lightly floured surface.
Spread tomato sauce over one half of each circle, then top with the sautéed mushrooms and mozzarella cheese.
Fold the other half of the dough over the filling and press the edges together to seal, ensuring the filling doesn't leak out.
Brush the top with beaten egg and prick a few holes with a fork to allow steam to escape during baking.
Bake in a preheated oven at 400°F (200°C) until golden brown, about 20 minutes.
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