Dissolve the yeast in a small bowl with a few tablespoons warm water and the sugar. Let sit 10 minutes.
Put the flour and salt in a large mixing bowl and make a well in the centre. Pour in the yeast mixture and olive oil. Stir with a wooden spoon, adding water one tablespoon at a time, until the dough pulls away from the sides of the bowl.
Transfer the dough to a floured surface and knead vigorously for at least 10 minutes. The dough is ready when it is smooth and elastic. Place in a lightly oiled bowl, cover it with a cloth and let it rise in a warm place at least 2 hours.
Preheat oven to 220 degrees C.
Divide the dough and roll it out into two pizzas. Place on pizza trays and spread with tomato pizza sauce and then sprinkle over half the mozzarella. Drizzle with olive oil and sprinkle with salt. Bake the pizzas, together or separately, 10 minutes.
Remove the pizzas from the oven and add the other half of the mozzarella and the mushrooms to each. Bake another 5 minutes. Remove from the oven again and add the prosciutto. Bake one last time no more than 3 minutes. Serve!