Preheat oven or grill with a pizza stone to 450 degrees F. On a lightly floured surface, roll out pizza dough to the largest thinnest rectangle you can make without the dough separating. Use a pizza cutter to divide the dough into four quarters. Place two of the dough pieces on the hot stone and cook for 5 minutes. They will puff up during cooking. Remove from oven and set aside with the side that was touching the stone facing up. Reduce heat to 350 degrees F.
Rinse the mushrooms in warm water and then cover them with one cup of boiling water and a lid. Allow to steep in the hot water at least 10 minutes.
Melt 2 tablespoons of the butter in a medium saucepan over medium heat. Cook the garlic until golden brown, about 2 minutes, taking care not to burn the garlic. Add the tablespoon of flour and allow to cook for a couple minutes while stirring. Add the steeping liquid from the mushrooms and 1 tablespoon of the minced rosemary and whisk together. Remove from heat and stir in half of the grated parmesan.
In a large bowl, combine 1 tablespoon of olive oil with thinly sliced potatoes and rehydrated mushrooms.
In a large heavy bottom melt the remaining 2 tablespoons of butter over high heat. Spread potato mushroom mixture to cover entire pan and allow to cook until the potatoes and mushrooms begin to brown, about 5 minutes. Flip with a spatula and cook for another 5 minutes. Return cooked potatoes and mushrooms to bowl and mix with remaining rosemary, remaining parmesan, salt and pepper.
Evenly spread the garlic mixture over each piece of flatbread and then top with mushroom potato mixture.
Heat in preheated 350 degree oven until bread gets golden brown and cheese has melted, about 15-20 minutes.
Slice and serve with an Estancia Pinot Noir for the perfect entertaining appetizer.