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1
Make the pizza dough by hand, in food processor, on in a stand mixer with the dough hook. Mix the flour and yeast, add the water and olive oil. The result should be a wet dough. Let sit for 20 minutes, then add in the salt. Place in a plastic bag and refrigerate overnight and for up to a week. Let come up to room temperature before using.
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2
Prepare the carmelized onions. Slice the onions thinly and cook in 2 Tbls butter and a pinch of salt over medium heat covered. After 10 minutes, uncover, lower the heat and cook, stirring often, until brown and sweet. Don't add sugar -- that's cheating. This can be done the day before. Refrigerate until ready to use.
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3
Slice the shitake caps into strips and cook gently in 2 Tbls butter with a little salt. Can be done the day before.Refrigerate until ready to use.
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4
Heat a gas grill or oven to 500 F. Divide the dough into four parts and roll out very thin on parchment paper using a little flour. Brush liberally with olive oil, sprinkle with salt and scatter onions, and mushrooms over, dot with cheese (not too much -- you should see the dough in underneath and through the toppings). Grind pepper over all.
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5
Use a pizza peel or cutting board to slide the pizzas into the hot oven or onto the grill -- parchment paper and all. If using a grill, cover. While the pizzas are cooking, shred the lemon basil. Bake for about 4 minutes, until the dough is cooked thoroughly and begins to brown. The parchment paper will blacken but not ignite. Once cooked, the pizzas will separate from the paper easily. Remove the pizzas and scatter the lemon basil over. Enjoy.
