Sift the all-purpose flour into a large bowl, add the dry yeast, salt, and sugar, then mix to combine.
Pour in the lukewarm water and olive oil, then start kneading. Work the mixture into a smooth, elastic dough, then cover and let it rise for 1 hour.
Clean and slice the mushrooms. Heat a little olive oil in a skillet, then add the minced garlic and mushrooms. Sauté for 5-6 minutes, until the mushrooms release their liquid and soften slightly.
Pour in the heavy cream, sprinkle with black pepper, and cook for another 2 minutes, until slightly thickened.
Divide the risen dough into two portions, then roll them out into circles on a floured surface.
Place the creamy mushroom mixture on one half of each dough circle, then sprinkle with grated mozzarella and Parmesan cheese.
Sprinkle with oregano, basil, and a pinch of black pepper.
Fold the other half of the dough over the filling, then crimp the edges to seal and prevent the filling from leaking 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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