In the bowl of a food processor, add flour, 1/2 tsp salt, and sugar. Briefly combine in food processor. Add cold butter pieces, and process until mixture resembles a coarse sand. Through the feeder tube, add water 1 tbs at a time just until dough adheres together. Wrap the dough in plastic wrap, and let rest in the refrigerator for a minimum of 1 hour. Meanwhile, in a large sauté pan over medium-high heat, melt 2 tbs butter, and sauté leeks, stirring occasionally and cooking until softened, about 8 minutes. Add remaining 1 tbs butter, mushrooms, and 1/4 tsp salt, and reduce heat to medium-low. Stirring occasionally, let mushrooms cook down until tender, about 20 minutes. Remove from heat, and reserve.
Preheat oven to 400°. Prepare a 12-inch rectangular quiche tart pan by greasing it with cooking spray. Remove dough from the refrigerator and place on a floured cutting board. Dust the top of the dough with flour, and use a rolling pin to evenly roll out the dough in a rectangle shape the best you can so it’s about 1 inch larger than the quiche pan. Roll the dough onto the rolling pin, and gently drop onto quiche pan. Press the dough into the quiche pan and up the sides of it, patching pieces together if needed. Use a fork to prick holes onto the dough, cover with foil, and line with pie weights, uncooked beans or a heat-proof dish, and cook in preheated oven for 8 minutes. Remove the weights and foil, and let pie crust cook 2 to 4 minutes more until slightly golden. Remove from oven, and reduce oven temp to 350°.
Using a spatula, spread the entire reserved mushroom-leek mixture into the cooked quiche crust in one even layer. In a medium bowl, whisk together eggs, milk, 1/4 tsp salt, and pepper. Pour egg mixture over the mushrooms. Slice the cheese into 1/2-inch slices, and place slices into 2 horizontal rows of 5 pieces each. Place in the oven, and cook until the center of the quiche is cooked, about 30 to 35 minutes. Let cool, and serve warm, cold or at room temperature.