Heat the oven to 550 degrees.
Melt 2 tablespoons of butter in a medium-sized skillet. Add the garlic cloves and ginger pieces to the pan once the butter is melted.
Place half of the mushrooms into the skillet. Sauté for 5 minutes over medium heat. Push the softened mushrooms to the perimeter of the skillet. Melt the remaining 1 tablespoon of butter in the center of the skillet.
Add the remaining 3 cups of mushrooms. Cook for an additional 5-7 minutes, or until the mushrooms are soft, slightly browned and sticking to the bottom of the pan. Stir occasionally.
Deglaze the pan with the wine. Increase the heat to medium-high and simmer for 3-5 minutes or until the liquid is absorbed. Remove from the heat. Remove the ginger and garlic.
Roll out the dough on a floured surface and place on your preferred baking equipment.
In a food processor, puree cream, vinegar, shallots and red pepper flakes until the cream is thickened and slightly fluffy.
Top the dough with the white sauce and sautéed mushrooms, parmesan, mozzarella. Bake for 10-12 minutes or until the dough and cheese are golden brown. Remove from the heat and let cool 5 minutes. Cut and serve immediately.