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1
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.
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2
Add 1 tablespoon of the oil, 1 teaspoon of the salt and the remaining 3/4 cup of water.
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3
Stir in as much of the flour as possible.
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4
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.
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5
Shape the dough into a ball and transfer to a lightly oiled bowl.
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6
Cover and let rise in a warm place until doubled in bulk, about 1 hour.
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7
Punch the dough down and shape it into 10 balls.
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8
Set the balls on a lightly oiled baking sheet, cover with lightly oiled plastic wrap and refrigerate for at least 1 hour or overnight.
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9
Heat the remaining 1/4 cup of oil in a large skillet.
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10
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.
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11
Preheat the oven to 550 or as hot as it will go.
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12
Put a pizza stone in the oven or generously brush 2 large baking sheets with oil.
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13
Working with 1 dough ball at a time, punch it down, then stretch or roll it out to a 6-inch circle.
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14
Sprinkle or spread some of the cheese on the dough and top with some of the mushrooms.
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15
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.
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16
Let cool slightly on a rack, then serve.