How to Make It
In a large bowl, dissolve the yeast and sugar in 1/4 cup of the lukewarm water and set aside until foamy, about 10 minutes. Add 1 tablespoon of the oil, 1 teaspoon of the salt and the remaining 3/4 cup of water. Stir in as much of the flour as possible. Turn the dough out onto a lightly floured surface and knead until silky, adding a little more flour if necessary; the dough should be soft but not sticky. Shape the dough into a ball and transfer to a lightly oiled bowl. Cover and let rise in a warm place until doubled in bulk, about 1 hour.
Punch the dough down and shape it into 10 balls. Set the balls on a lightly oiled baking sheet, cover with lightly oiled plastic wrap and refrigerate for at least 1 hour or overnight.
Heat the remaining 1/4 cup of oil in a large skillet. Add the mushrooms and shallot, season with salt and pepper and cook until the liquid evaporates and the mushrooms begin to brown, about 9 minutes.
Preheat the oven to 550° or as hot as it will go. Put a pizza stone in the oven or generously brush 2 large baking sheets with oil. Working with 1 dough ball at a time, punch it down, then stretch or roll it out to a 6-inch circle. Sprinkle or spread some of the cheese on the dough and top with some of the mushrooms.
Working in batches, bake the pizzas until the crusts are golden and the bottoms are crisp, about 7 minutes on a pizza stone or 11 minutes on a baking sheet. Let cool slightly on a rack, then serve.