Preheat oven to 500 degrees with rack in lowest position and a pizza stone on rack. Melt butter in a large saute pan over medium-high heat. Add shallots; cook until softened, about 2 minutes. Add mushrooms, 1 1/2 teaspoons salt, and 1/4 teaspoon pepper; cook until mushrooms are softened and golden brown, 5 to 7 minutes. Add wine, bring to a boil, and cook until absorbed, about 1 minute. Add cream; cook, stirring, until mostly absorbed, about 4 minutes. Transfer to a plate, and let cool.
Gently stretch each piece of dough into a 6-by-16-inch rectangle, and place on parchment. Brush dough with oil, and spread with mushrooms, leaving a 1/2-inch border. Transfer pizzas on parchment onto pizza stone, and bake until crust is golden and crisp, 10 to 12 minutes. Drizzle with oil, sprinkle with tarragon, and cut into wedges.