** MAKE DOUGH** Mix the yeast, salt, sugar and olive oil with the water in a 5-quart bowl or a lidded (not air-tight) food container.
Mix in the flour without kneading using a spoon, a large capacity food processor (with dough attachment), or a heavy-duty stand mixer (with dough hook). If you're not using a machine, you may need to use wet hands to incorporate the last bit of flour. I use a spoon to combine, then my hands for the last bit of flour.
Cover (not air-tight) and allow to rest at room temperature until dough rises and collapses (or flattens on top), approximately 2 hours.
The dough can be used immediately after the initial rise, though it is easier to handle when cold. Refrigerate in a lidded (again, not air-tight) container and use over the next 12 days.
** MAKE PIZZA ** Melt 1 tablespoon butter with 1 tablespoons vegetable oil in heavy large skillet over medium heat. Add onions and sauté until golden, about 45 minutes. Season with salt and pepper.
Melt remaining 2 tablespoons butter with 1 teaspoon vegetable oil in another heavy large skillet over medium-high heat. Add mushrooms, garlic, and shallot. Sauté 4 minutes. Add wine and simmer until almost all liquid is absorbed, stirring frequently, about 13 minutes. Add rosemary; season with salt and pepper.
Position rack in bottom third of oven. Place a pizza stone on rack. Preheat oven to 500°F at least 30 minutes before baking.
Instead of kneading this dough, you're going to "cloak" it, which (and this is VERY important), should take no more than 30 seconds for the whole process from grabbing off a hunk of dough to the end of cloaking. This is done after the 2-hour rest. Sprinkle a little flour on the surface of the dough you've made and pull or cut off a grapefruit-sized ball. This will be for one pizza. Sprinkle a bit more flour on the top of that ball and with the entire length of the sides of your thumbs, gently stretch the top surface around to the under part of the ball. Gently! You're not actually tearing the dough, but think of it as lengthening the fibers on the very top surface of the ball by stretching it just a bit.
Roll out 2 dough disks on lightly floured surface to 10-inch rounds, allowing dough to rest a few minutes if it springs back. Sprinkle 2 pizza peels with cornmeal and transfer 1 dough round to each peel. Lightly brush dough with garlic oil. Sprinkle one with 1/2 cup cheese, and then scatter 2 1/2 tablespoons onions over cheese. Scatter 1/2 cup mushrooms over onions. Sprinkle with salt and repeat for second pizza.
Position tip of pizza peel at a slightly downward angle to roughly 2/3 of the way across pizza stone. With quick forward jerks and less jerky backward motions, jiggle the pizza from the peel to the stone, then slide peel the remainder of the way out so that the pizza is centered on the stone. Repeat with second pizza and bake for 6 minutes. Rotate pizzas half a turn. Bake until crust is deep brown, about 6 minutes longer, but watch that the crust doesn't burn.
Using a large spatula or two, carefully transfer pizzas to cutting board. Let rest 1 minute. Slice into wedges and serve.