Step 1

Preheat oven to 500 degrees with rack in lowest position. Melt butter in a large skillet 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 golden brown, 5 to 7 minutes. Add wine, bring to a boil, and cook until absorbed, about 1 minute. Add cream and stir to combine. Transfer to a plate, and let cool.

Step 2

Working with one piece of dough at a time, gently stretch into a 9-inch circle and place in an 8-inch cast-iron skillet. Brush dough all over with olive oil, and sprinkle with 1-ounce cheese, and one-quarter of the mushrooms. Top with another ounce of cheese. Drizzle with oil, and season with salt and pepper.

Step 3

Place skillet in the oven and bake until crust is golden brown, 15 to 16 minutes. Sprinkle with tarragon and cut into wedges. Serve immediately. Repeat process with remaining ingredients.