For the dough: Mix 500 g flour, yeast, 1⁄2 TL sugar and salt in a mixing bowl. 1⁄4 Add lukewarm water, knead everything smooth with the dough hooks of the mixer. Cover the dough and let it rise in a warm place for about 1 hour.
For the sauce: Peel and chop the onion and garlic. Heat 1 tablespoon of oil. Fry the onion and garlic in it. Add the tomatoes, chopping them with a spatula. Season with basil, salt, pepper and 1 pinch of sugar. Simmer open for about 10 minutes. Season to taste.
Clean, wash and quarter the mushrooms. Heat 2 tablespoons of oil in a pan. Fry the mushrooms in it. Season with salt and pepper.
Knead the dough and roll out into 4 portions on a little flour (approx. 20 cm Ø). Spread each portion with 1⁄4 sauce, leaving about 2 cm of edge free. Cover with ham and spread mushrooms on top. Spread the edges of the dough with water and fold over half of each portion. Press on the edges. Sprinkle with cheese on a baking tray lined with baking paper. Bake in a hot oven (electric cooker: 200 °C/circulating air: 175 °C/gas: see manufacturer) for about 25 minutes.