Prepare the Crust: Pulse flour and salt in a food processor until well combined. Add butter and goat cheese, and pulse until mixture resembles coarse meal, about 12 times. With processor running, add egg yolk and ice water through the food chute, processing just until mixture forms a ball, about 30 seconds. (Dough will be sticky.) Shape dough into a flat disk, wrap in plastic wrap, and chill 2 hours or up to 3 days.

Preheat oven to 400°F. Roll dough to 1/8-inch thickness on a lightly floured surface. Using an inverted bowl as a guide, cut out 6 (6-inch) rounds. Gently press each dough round into a 4 1/2-inch fluted tart pan with removable rim; freeze 15 minutes.

Line each tart shell with parchment paper, and fill with pie weights. Bake in preheated oven until just set, 12 to 15 minutes. Remove weights and parchment paper, and prick pastry on bottoms of shells with a fork. Return tart shells to oven, and bake until pastry is light golden brown, 8 to 10 minutes. Transfer tart pans to a wire rack, and cool completely, about 20 minutes. Reduce oven temperature to 350°F.

Prepare the Filling: Cut 2-inch pieces from asparagus tip ends, and reserve. Cut stalks into 1/2-inch pieces. Melt butter in a skillet over medium-high. Add leek, and cook, stirring, until tender, about 5 minutes. Stir in asparagus stalk pieces, and cook, stirring, until bright green, about 1 minute. Remove from heat, and cool 5 minutes.

Whisk together eggs, cream, parsley, salt, and pepper until combined. Transfer tart shells to a baking sheet. Divide leek mixture, goat cheese, and egg mixture evenly among tart shells. Place 2 or 3 of the reserved asparagus tips on each quiche.

Bake at 350°F until just set in the middle, 22 to 24 minutes, covering with foil after 15 minutes, if needed, to prevent excess browning. Transfer to a wire rack; cool at least 15 minutes. Remove tarts from pans.