Thinly sliced russet potatoes absorb the richly flavored custard as it bakes and give this quiche its layered texture.
For the buttery flaky pastry:
In a food processor, pulse flour and salt 3 or 4 times or until combined. Add butter, and pulse 5 or 6 times or until mixture resembles coarse meal. With processor running, gradually add 1/4 cup ice-cold water, and process just until dough forms a ball and pulls away from sides of bowl, adding up to 1/4 cup more ice-cold water if needed.
Gently form dough into a flat disk; wrap in plastic wrap, and chill 1 hour to 2 days.
Preheat oven to 400°F. Roll dough into a 12-inch circle (about 1/8 inch thick) on a lightly floured surface. Fit into a lightly greased 9-inch round tart pan with removable bottom, pressing dough on bottom and up sides into fluted edges.
Line dough with parchment paper, and fill with pie weights or dried beans.
Bake for 20 minutes. Remove weights and parchment paper, and bake 15 to 20 more minutes or until bottom is golden. Transfer to a wire rack. Cool completely (about 30 minutes).
For the filling:
Preheat oven 350°F.
Bring potatoes, 1 teaspoon salt and water to cover to a boil in a large skillet over medium-high heat; cook 5 to 7 minutes or until tender. Drain potatoes, and pat dry with paper towels. Cool 10 minutes.
Cook onions in boiling salted water to cover in a small saucepan 2 to 3 minutes or until tender. Drain well, and press between paper towels.
Stir together parsley and next 2 ingredients in a small bowl.
Whisk together half-and-half, next 4 ingredients, half of parsley mixture, and the remaining 1/2 teaspoon salt.
To assemble and bake:
Spread half of potatoes in bottom of pastry crust. Top with half of Gruyère cheese, half of cooked onions, and half of egg mixture; repeat layers once. Sprinkle with Parmesan cheese.
Bake on an aluminum foil-lined baking sheet 40 to 45 minutes or until center is set. Let stand 10 minutes. Sprinkle with remaining parsley mixture. Serve warm or at room temperature.