Fry the onions in the butter in a large skillet over medium-low heat, stirring occasionally, until the onions are deep golden brown, 15 to 20 minutes (maybe longer). Set aside to cool.
Preheat the oven to 400 degrees F.
Bring a pot of water to a boil and salt generously; prepare an ice water bath. Blanch the broccoli in the boiling water for 1 minute, then remove it to the bowl of ice water to stop the cooking and set the color. Remove it from the water to drain.
Roll out the pie crust on a floured surface and press into a deep fluted tart pan with a removable bottom (or a deep-dish pie pan). Lay the onions over the base of the crust, then lay the brocolli pieces on top of the onions. Chop the Brie and lay that in the crust too.
Whip the half-and-half, eggs and some salt and pepper in a large bowl, then pour it into the crust. Sprinkle the chives on top.
Place the pan on a rimmed baking sheet, cover lightly with aluminum foil and bake for 40 to 45 minutes. Remove the foil and continue baking until the quiche is set and the crust is golden brown, another 10 to 15 minutes. (The quiche will still seem slightly loose, but will continue to set once removed from the oven.) Let the quiche sit for 10 to 15 minutes, then, if using a tart pan, remove the quiche from the pan. Cut into slices with a sharp serrated knife and serve.
Put the flour in a large bowl. With a pastry cutter, gradually work the shortening and butter into the flour until it resembles coarse meal, 3 to 4 minutes. In a small bowl, beat the egg with a fork and pour it into the flour mixture. Add the cold water, white vinegar and salt. Stir gently to combine.
Form the dough into 2 evenly sized balls and place each ball into a gallon food storage bag. Using a rolling pin, slightly flatten each ball of dough (to about 1/2 inch thick) to make rolling easier later. Seal the bags and place them in the freezer until you need them. (If you will be using them immediately, it's still a good idea to put them in the freezer for 15 to 20 minutes to chill.) When you are ready to use the dough to make a crust, remove from the freezer and allow to thaw for 15 minutes.