For the dough, combine the flour, yeast, salt, and sugar in a large bowl. Add the warm water and olive oil, and knead until you have a smooth, elastic dough.
Cover the dough with a clean kitchen towel and let it rise for 1 hour, or until doubled in size.
Meanwhile, prepare the filling: chop the mushrooms and ham, then mix with the tomato sauce, minced garlic, and oregano.
Preheat the oven to 430°F (220°C).
Divide the risen dough into two portions and roll each out into a circle, about 10 inches (25 cm) in diameter.
Spoon the filling onto one half of each circle, sprinkle with the grated mozzarella, then fold the other half over to create a half-moon shape. Press the edges firmly to seal.
Brush the tops of the calzones with beaten egg and place them on a baking sheet lined with parchment paper.
Bake in the preheated oven for 15-20 minutes, or until golden brown.
Let cool for a few minutes before serving warm.
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