Step 1

Heat oven to 375°F. Melt butter in large skillet on medium. Add onions, 1 teaspoon salt, and 1/2 teaspoon pepper and cook, covered, stirring occasionally, 8 minutes. Uncover and cook, stirring occasionally, until beginning to turn golden brown, 4 to 6 minutes. Remove from heat and fold in thyme and parsley.

Step 2

Let disk of dough soften slightly at room temperature, 5 to 10 minutes. On lightly floured surface, with floured rolling pin, roll out to about 12-inch round and fit into 9-inch pie plate, folding overhang under and pinching dough to form flat edge that extends to outer edge of pie plate lip. Refrigerate until ready to use.

Step 3

On lightly floured sheet of parchment, with floured rolling pin, roll out rectangle of dough to about 7- by 13-inch rectangle, about 1/8 inch thick. Slide onto baking sheet and freeze until firm, 10 minutes. Using ruler and pastry wheel, cut into 1/4- by 7-inch strips. Working with 3 strips at a time (keep remaining dough chilled), pinch together ends of strips and braid as tightly as possible. Refrigerate braid and repeat to make 5 to 6 braids total.

Step 4

Beat 1 egg with 1 teaspoon water and lightly brush border of chilled pie shell; reserve egg wash. Press braids firmly around outer edge of border to adhere, trimming and fitting to create appearance of 1 seamless braid. Prick bottom and sides of pie shell lightly with fork and freeze 15 minutes.

Step 5

Transfer pie plate to rimmed baking sheet. Press large piece of parchment paper into pie shell and fill with pie weights. Bake 20 minutes. (If braid slips during baking, prop up with small ball of foil.) Remove parchment and weights and bake until bottom is set, about 10 minutes. Brush inside with egg white and bake until dry, about 5 minutes.

Step 6

In large bowl, whisk together remaining eggs, milk, sour cream, nutmeg, cayenne, and 1/4 teaspoon salt; fold in onions and cheese. Pour into pie shell.

Step 7

Bake, tenting edges with foil if they get too dark, until quiche is golden brown and just set, 45 to 55 minutes. Let rest at least 10 minutes before serving.