Heat the oven to 375 degrees F and prepare a baking sheet with a layer of aluminum foil. Spread the prepared piecrust in a 9-inch pie pan and crimp the edges of the dough. Chill the crust in the freezer while you prepare the other ingredients.
Melt the butter in a non-stick skillet over medium heat, and sauté the spinach until just softened. Drain the spinach and set aside, making sure to squeeze out any excess liquid.
Meanwhile, spread the asparagus out on the baking sheet and roast for 10 minutes, until bright green and just tender.
In a large bowl, whisk together the eggs, milk, salt, black pepper, half of the fresh thyme, and all of the goat cheese to make a custard base.
Remove the chilled pie crust from the freezer. Spread the drained spinach over the bottom crust, and pour the egg custard over top. Sort the asparagus spears by size. Gently lay the spears into the custard so that they form lines, with the longest in the center. Sprinkle with the remaining fresh thyme.
Bake the quiche for 50 minutes to 1 hour, until the custard is set and the crust lightly browned. Remove the quiche from the oven and let cool for 30 minutes before serving.