Preheat the oven to 350°F.
Cut the chilled unsalted butter (⅓ cup) into small pieces. In a food processor combine the flour (1½ cups), chopped fresh thyme (1 teaspoon), ⅛ teaspoon salt, and ⅛ teaspoon freshly ground black pepper. Add the butter and pulse several times until the mixture looks like coarse crumbs with pea-sized pieces of butter still visible.
In a small cup stir together the ice water (1½ tablespoons) and white vinegar (1 teaspoon). With the processor running, slowly add the water–vinegar mixture, pulsing just until the dough begins to come together. If needed, add a little more ice water a teaspoon at a time. Stop when the dough holds when pinched but is not wet or sticky.
Turn the dough out onto a piece of plastic wrap, form it into a ball, then flatten it into a disc about 4 inches across. Wrap and refrigerate for 15 minutes.
While the dough chills, make the filling: Heat the extra-virgin olive oil (1 teaspoon) in a sauté pan over medium heat. Add the diced red onion (¼ medium) and minced garlic (1 large clove) and sauté, stirring often, for 2 minutes. Add the mushrooms (4 ounces, cut into ½-inch pieces) and cook, stirring occasionally, until the mushrooms are soft and the pan is dry of liquid, about 5 minutes. Remove from heat and let cool slightly.
On a lightly floured surface, roll the chilled dough between the two sheets of plastic wrap to a circle large enough to line a 9-inch pie plate. Peel off the top sheet of plastic, invert or transfer the crust into the pie plate, then peel off the remaining sheet of plastic. Gently press the crust into the plate and trim the edges; flute the edges if you like. (At this point the crust may be refrigerated for up to 1 day or frozen up to 1 month.)
In a medium bowl whisk the eggs (8 large) until blended. Stir in the chopped fresh basil (6 leaves), chopped fresh oregano (2 teaspoons), freshly grated nutmeg (1/8 teaspoon), white pepper (1/4 teaspoon), sea salt (1/2 teaspoon), and freshly ground black pepper (1/8 teaspoon) until evenly combined.
Place the pie plate on a baking sheet. Scatter the crumbled goat cheese (4 ounces) evenly in the crust, then spread the sautéed mushrooms and onion mixture over the cheese.
Pour the egg and herb mixture evenly over the filling in the crust.
Bake on the baking sheet for 35 to 45 minutes, or until the quiche is slightly puffed, springy to the touch, and a toothpick inserted into the center comes out clean.
Remove from the oven and let the quiche rest for about 10 minutes before slicing and serving.