Pulse flour, butter, granulated sugar, and powdered sugar in a food processor until combined and sand-like in texture, 12 to 15 pulses. Add whole egg; pulse until a dough forms, about 12 pulses. Divide dough in half, and shape each half into a 6-inch disk; wrap each half in plastic wrap. Chill 1 portion of dough until firm, about 30 minutes; reserve remaining dough in freezer for another use.
Preheat oven to 375°F with racks in center and bottom third positions. Unwrap chilled dough on a lightly floured work surface, pressing out dough slightly using the palm of your hand. Roll dough into a 14-inch round, about 1/4-inch thick. Transfer dough to a 9 1/2-inch tart pan with a removable bottom, pressing dough into bottom and up sides of pan. Tuck dough into corners of tart pan to prevent air pockets; trim dough overhang around edges. Prick bottom of dough all over using a fork. Chill in refrigerator for 30 minutes.
Line bottom of dough using parchment paper; fill with pie weights or dried beans. Place on a large baking sheet; bake on center rack until Pastry is set and golden brown on edges, about 25 minutes. Remove from oven; carefully remove parchment paper and weights. Reduce oven temperature to 325°F. Lightly brush beaten egg yolk on bottom of Pastry. Return to oven, and bake at 325°F until inside of Pastry is dry, 8 to 10 minutes. Remove from oven, and let cool for 10 minutes.
While Pastry is baking, stir together heavy whipping cream, brown sugar, and miso in a large saucepan; cook over medium, stirring often, until sugar is dissolved, 2 to 3 minutes. (Do not boil.) Remove from heat, and let cool until just warm to the touch, about 10 minutes.
Cook pumpkin in a small saucepan over medium, stirring constantly, until slightly thickened, about 5 minutes (you should now have 1/4 cup). Set aside, and let cool, about 5 minutes. Whisk egg and egg yolk in a large bowl until combined. Slowly pour cooled cream mixture into eggs in a slow steady stream, whisking until incorporated. Whisk in pumpkin until combined. Slowly pour pumpkin mixture through a fine-mesh strainer into prepared Pastry, pushing any undissolved miso using the back of a spoon against strainer.
Bake at 325°F in bottom third of oven until slightly puffy on edges and lightly jiggly in center, 30 to 35 minutes. (Tart is cooked when it wobbles like homemade jelly in the center; custard will set once cooled.) Remove from oven, and let cool completely, about 1 hour, before serving.
