Preheat the oven to 475°.
Let the pizza dough sit at room temperature (this is called proofing) for about 30 minutes in a large bowl coated with cooking spray.
In a large skillet over medium-high heat, add the butter and let it melt. Add the mushrooms, the shallot, the garlic, and 1/4 teaspoon of the salt.
Cook, stirring frequently, for about 15 minutes, or until most of the moisture has been evaporated from the mushrooms. It should be a deep, dark-colored paste. Allow to cool to room temperature.
Heat another large skillet over medium heat. Spray with cooking spray.
Add the onions and sweat for 20 minutes or until soft and translucent.
Add the riesling, raise the heat to medium-high, and stir 30 seconds or until the wine has fully evaporated.
Fold in the rosemary. Season with 1/4 teaspoon salt and pepper.
Stretch the pizza dough quite thinly, as to make a crackerbread, until it is about 1/8 inch to 1/4 inch thick.
Lay the dough on a perforated pizza pan. If you don't have one, use a baking sheet. The crust will be a tad less crisp.
Spread the mushroom mixture in an even layer on the dough, leaving a 1-inch border. Top evenly with the onions, and finish with the cheese.
Bake at 475° for 15 minutes, or until the dough is completely crisp.