Combine 1 cup of the oil and the garlic in a small pot over medium-high heat and bring to a simmer. Turn the heat to low and cook until the edges of the garlic are slightly golden brown, 20 to 25 minutes. Turn off the heat and let cool.
Puree the garlic, 1/2 cup of the cooking oil and 1/2 teaspoon kosher salt until smooth. Set aside. (The garlic puree will keep in an airtight container in the refrigerator up to 1 month.)
Heat the remaining 2 tablespoons of oil in a large cast-iron skillet over high heat. Saute the mushrooms in batches (do not overcrowd the pan). Add another tablespoon of oil between batches if needed. Do not stir the mushrooms until they brown on one side, about 3 minutes. Add some of the thyme, then stir the mushrooms and cook for another 2 minutes. Once all the mushrooms have been browned, combine them in the skillet and season with 1 teaspoon kosher salt and 1/2 teaspoon pepper. Add the wine and cook until evaporated, about 3 minutes. Add the creme fraiche and stir until fully incorporated. Turn off the heat and keep warm.
Prepare a grill for medium-high heat. Roll out the pizza dough on a floured cutting board to about 1/2-inch thick. Lay the dough on the grill and cook until grill marks form, 4 to 5 minutes. Flip and cook for 3 to 4 minutes more. Spread with 2 to 3 tablespoons of the garlic puree. Cover the grill and cook for 30 seconds. Transfer the flatbread to a cutting board and top with the mushrooms and the parsley. Sprinkle with 1/2 teaspoon flaky sea salt.