In a mixing bowl, combine the chilled unsalted butter, all-purpose flour, chopped fresh thyme, salt, and freshly ground black pepper. Use a pastry cutter or your fingers to blend until the mixture resembles coarse crumbs.
Add the ice water and white vinegar to the mixture. Stir until the dough begins to come together. If necessary, add more ice water, a tablespoon at a time, until the dough is cohesive.
Form the dough into a disk, wrap it in plastic wrap, and refrigerate for at least 30 minutes. This helps the crust maintain its shape during baking.
While the dough chills, heat the extra-virgin olive oil in a skillet over medium heat. Add the diced red onion and minced garlic, and sauté until the onion is translucent. Add the mixed mushrooms and cook until they are softened and any liquid has evaporated. Season with salt and pepper to taste. Remove from heat and let cool.
Preheat your oven to 375°F (190°C).
Once the dough has chilled, place it on a lightly floured surface. Roll it out into a circle about 12 inches in diameter. Carefully transfer the dough to your pie dish, pressing it into the bottom and up the sides. Trim any excess dough.
In a large mixing bowl, whisk together the eggs, chopped fresh basil, chopped oregano, freshly grated nutmeg, white pepper, freshly ground black pepper, and sea salt. Stir in the sautéed vegetables and crumbled goat cheese until evenly distributed.
Pour the filling into the prepared crust, spreading it evenly.
Place the quiche in the preheated oven and bake for 35-40 minutes, or until the filling is set and the top is lightly golden.
Remove the quiche from the oven and allow it to cool for about 10 minutes before slicing. Serve warm or at room temperature.