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Preparation Roll dough to a 15" round on a lightly floured work surface; trim edges. Fit dough into a 9" springform pan, letting dough fold over itself slightly around the edges so that it creates a gently pleated, rippled effect (the dough will extend above top of pan). Place pan on a rimmed baking sheet. Chill 1 hour, or freeze 20 minutes.
Preheat oven to 350°F. Dock bottom surface of dough with a fork. Line dough with parchment paper or foil, then fill with dried beans or baking weights. Bake crust on baking sheet until top edge is golden, about 30 minutes. Using parchment, lift out beans, then continue to bake until surface of crust is set and lightly browned and feels dry, 35–40 minutes more. Transfer pan to a wire rack and let cool.
While crust cools, reduce oven temperature to 325°F. Cook shallots, oil, and 1/2 tsp. salt in a small skillet over medium-low heat, stirring often, until tender and translucent, about 5 minutes. Remove from heat and stir in chives. Spoon filling into cooled crust.
Vigorously whisk eggs in a large bowl or glass measuring cup. Whisk in half-and-half, pepper, 1 cup crème fraîche, and remaining 1 tsp. salt until combined. Pour custard over filling. Bake quiche until filling is puffed, set, and golden, 60–75 minutes. Transfer pan to a wire rack and let cool at least 1 hour. Release springform, then lift quiche off base using parchment overhang and transfer to a platter or cutting board.
Serve with braised vegetables, prosciutto, crème fraîche, green goddess dressing, avocado, greens, and lemon wedges alongside for guests to dress their quiche as desired.
Do Ahead: Crust can be baked 1 day ahead. Store uncovered at room temperature. Quiche can be made 6 hours ahead. Store uncovered at room temperature. Quiche can also be made 1 day ahead; cover and chill overnight. Bring to room temperature 2 hours before serving.