Combine flour and 1/2 teaspoon salt; cut in butter and shortening with pastry blender until mixture is crumbly. Sprinkle water (1 tablespoon at a time) over surface, stirring with a fork until dry ingredients are moistened. Shape into a ball; chill 1 hour.
Roll pastry to a 12-inch circle on a floured surface. Line a 10-inch quiche dish with pastry; trim excess pastry around edges. Chill 30 minutes. Line dish with aluminum foil; add pie weights. Spread pie weights over bottom and up sides of foil-lined pastry. Bake at 425° for 15 minutes. Remove aluminum foil and pie weights. Bake an additional 5 minutes. Let cool completely on a wire rack.
Saute onion, leeks, and garlic in 2 tablespoons butter in a large skillet over medium-high heat until vegetables are tender. Remove from heat; stir in tarragon. Sprinkle 1/4 cup cheese in baked pastry shell. Spoon onion mixture over cheese. Combine eggs and remaining ingredients; stir with a wire whisk. Pour over onion mixture in pastry shell. Sprinkle with remaining 3/4 cup cheese. Bake at 350° for 30 minutes or just until set. Let stand 10 minutes before slicing.