Dissolve the yeast and the sugar in a few tablespoons of warm water, pour all the flour in a large bowl or directly on your work surface and pour the yeast mixture in the middle of the flour. Mix together the warm water, the oil, the salt and then little by little pour them in the middle of the flour mound. Then knead until you get a nice dough.
Tuck in the ends of the dough and shape it into a ball, smooth and compact, then turn it upside down, and roll it between your hands with the palms facing upwards in a V shape. This step is called “pirlatura”.
Put the dough in an oiled bowl and cover with a dishcloth, then let it rise for about 3 hours in a warm place. Cut the dough in 4 parts, shape them into balls, handling them as little as possible, then let them rest on a baking tray, covered and in a warm place.
In the meantime heat the oil with the garlic in a non-stick frying pan, chop half of the parsley and pour in the pan the cleaned and roughly chopped mushrooms, then sauté them for a few minutes, If using frozen mushrooms, defrost them completely and pat them dry with kitchen paper.
Set the cooked mushrooms aside and in the same pan sauteé the chestnuts, crushing half of them with the back of a spoon.
When the dough has completely proofed, roll out each ball on a work surface sprinkled with durum wheat semolina, spreading the dough outwards, using your fingers and without touching the edges, in order to create while baking the classic tall border.
Top the pizzas with the sautéed mushrooms, add the cheese, cut in very thin slivers, and the chestnuts.
Bake in a 220° C oven for 18 minutes on the lowest rack, then about 2 more minutes on the highest rack to get a slightly charred crust on the borders.
When you serve it add some prosciutto slices and some parsley.