Heat oven to 400°. Line two baking sheets with parchment. Combine ¼ cup oil and garlic in a small bowl. Brush flatbread with garlic oil. Evenly distribute parmesan between flatbreads. Arrange asparagus in a crosshatch pattern on top. Brush with additional oil and season with salt and pepper. Bake for 10 minutes.
Meanwhile, heat remaining 2 tablespoons oil in a small skillet over medium heat. Add turmeric to oil and then eggs. Season with salt and pepper. Cook until edges are crisp and whites are mostly set.
Remove flatbreads from oven and top each with a fried egg and goat cheese. Return to oven until tops of egg whites are cooked and goat cheese is warmed through, 3–5 minutes.
NOTE: Although Sami’s Bakery Millet & Flax Lavash is not certified gluten free, it is made without wheat or any ingredients that contain gluten. You can purchase Sami’s Bakery products at Earth Origins and Simon’s Coffee House.