Heat oven to 475ºF. If you have a pizza stone or cast iron skillet, place it in the oven now.
Heat a large skillet over medium heat. Add the shallots and cook until they begin to stick, 2-3 minutes. Add the garlic and mushrooms. Stir well until the mushrooms also begin to stick, but do not let them burn. You should see some color on the bottom of the pan.
Add the wine and stir to release the browned bits from the pan. Sauté until the mushrooms have released their liquid and the pan is almost dry, about 4 minutes. Stir in the chives. Set aside.
To make the crust, combine the flour, cornstarch, sugar, baking powder, and salt in a large mixing bowl. Stir in the water and 2 tablespoons olive oil. Knead a few times, until the dough comes together. Split in half.
On a floured surface, roll half of the dough into a 9-inch circle. Brush with the remaining olive oil. Top with half the cheese and half of the mushrooms. Repeat with remaining ingredients.
Cook the pizzas on the preheated stone or skillet (if using) or on a heavy baking sheet for 9-11 minutes, or until the crust is crispy and the cheese is melted.
Let the pizzas cool for 2-3 minutes, then cut into 8 slices.