Preheat oven to 450 degrees. Shape the pie crust into a 9-inch tart pan. Place the tart pan on a sheet pan and blind bake for 10-12 minutes until golden brown. Reduce oven to 400 degrees.
Clean the leeks: trim the root end and the dark green leaves at the top. Slice in half lengthwise and then slice crosswise into ½-inch slices. Add the leeks to a salad spinner. Rinse well and spin dry.
In a large skillet, melt the butter and add the leeks. Season with salt and pepper and thyme leaves. Saute over medium heat until soft and lightly browned, about 10 minutes. Set aside.
Blanch the asparagus by adding the chopped asparagus to a pot of boiling water. Cook for 30-60 seconds. Transfer the asparagus to a bowl of ice water to stop the cooking. Drain and set aside.
In a mixing bowl, whisk together the eggs and heavy cream. Season with ¼ teaspoon of black pepper.
Spread the leeks evenly over the pie crust. Top with half of the Brie. Arrange the asparagus over the Brie and pour the egg mixture over the top. Arrange the remaining Brie over the top of the quiche. Bake with a sheet pan under the tart pan to catch any spills.
Place the tart pan or pie dish on a sheet pan. Bake at 400 degrees for 30 minutes or until the custard is set in the middle. Let the quiche rest for 5-10 minutes before slicing.