Multi-Mushroom Flatbread Pizza

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Dough:

1 1/2 cups warm water, between 100 and 115 degrees F, as measured with a candy thermometer (any hotter will kill the yeast - an organic leavener, too cool and the yeast won't be activated)

1 tablespoon salt

1 tablespoon sugar

One 1/4-ounce packet yeast

4 cups all-purpose flour, plus some extra for the dough board

1/4 cup cornmeal, for pizza stones

Base topping:

2 tablespoons butter

3 shallots, sliced

3 large sprigs fresh thyme, leaves stripped from the stems and minced

3 ounces dried porcini mushrooms, soaked in hot water to re-hydrate and remove grit (24 ounces if using fresh)

Salt and pepper

1/2 cup heavy cream

Topping:

2 tablespoons extra-virgin olive oil

3 ounces fresh porcini mushrooms, cleaned and trimmed

3 ounces fresh cremini mushrooms, cleaned and trimmed

3 ounces fresh shiitake mushrooms, cleaned and trimmed

2 cloves garlic, lightly crushed with the side of a knife blade, then minced

3 large sprigs fresh thyme leaves, stripped from the stems

1/2 pound fresh mozzarella cheese, sliced