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Dissolve salt, sugar, and yeast in the warm water and allow the yeast to proof. Proofing takes about 15 minutes. Place flour in a food processor fitted with a dough blade, and through the feed tube with the food processor running, slowly pour in the proofed yeast mixture, until the dough comes together and is a cohesive mass.
Transfer the dough to a floured board, and knead for about 5 minutes. Place in a bowl, cover with a clean dish towel and allow dough to rise, so that it roughly doubles in volume. (This will take about 30 minutes to 1 hour). The dough has risen enough if you make an indentation with your finger and it does not spring back.) Punch the dough down and divide into 6 equally sized balls. Cover with towel and allow it to rise again. (Allowing the dough to rise a second time gives it a finer texture.) Note: It will not rise as much the second time.
While the dough is rising, prepare the "base" pizza topping. Heat the butter in a large saute pan over medium-high heat and saute shallots and thyme until the onion begins to turn translucent. (If you are using dried mushrooms, remove them from their soaking liquid, lifting them out so all grit remains at the bottom.) Add the porcini to the pan, season with salt and pepper, to taste, and cook until tender. Remove from heat, stir in heavy cream, and scoop into a food processor. Pulse to chop and set aside to rest. This is your base topping
Preheat oven to 450 degrees F. On a floured surface, stretch the dough into 6 (6 by 8-inch) ovals and shake cornmeal over both sides of the dough. (The crust will be par-baked on both sides on the stovetop before you add the topping to be baked.) In a heavy bottomed 12-inch skillet which has a lid, heat a small amount of oil over medium heat. (Swirl the oil to coat the entire inside of the skillet.) Transfer each dough oval to the skillet, allowing 1 side to lightly cook and stiffen up before flipping it over with tongs. When both sides of each oval are par-baked this way, transfer to a pizza stone. Spoon the base topping into the center of the now toasted crusts, spreading evenly over the surface.
For the finishing topping, heat the olive oil over medium heat in a large saute pan and cook the porcini, cremini, shiitake, garlic and thyme until tender. Spread evenly over the base topping and lay fresh mozzarella slices over.
Bake in the oven for about 10 minutes until cheese melts and edges of crust are golden.