Heat oven to 375° (190ºC) with rack set in middle position.
Roll dough out into an 11-inch, ¼-inch thick, round. Transfer dough to a 9-inch deep dish pie plate and gently press into bottom and up sides. Fold edges under and crimp, freeze 20 minutes. Line pie shell with parchment paper and fill with baking weights or beans.
Bake crust until edges are golden brown, and bottom is set 40–45 minutes. Remove parchment and baking weights, and bake until bottom is golden brown about 15 minutes more.
Meanwhile, cook onions in oil over medium heat in a nonstick skillet until softened and starting to brown, 5–7 minutes; let cool slightly.
For the custard, whisk together eggs, milk, salt, pepper, paprika and nutmeg until thoroughly combined; stir in onions.
Combine broccoli, Cheddar and Muenster then sprinkle into bottom of hot crust.
Pour custard over top.
Bake quiche until edges are set and center is slightly jiggly, 1–1½ hours (if crust edges start to get too brown, use pieces of foil to shield them).
Let cool briefly then serve warm or at room temperature.