Heat the oven to 350°F and arrange a rack in the middle. Lightly flour a work surface and roll pie dough out into a 12-inch round approximately 1/4 inch thick. Line a 9-inch pie plate with the dough, trimming within 1 inch of the plate. Fold over excess dough and pinch to create a decorative edge.
Line the crust with a piece of parchment paper or aluminum foil large enough to overhang the edge by 1 inch, then fill with pie weights or dried beans. Bake the crust on a baking sheet until set and light brown, about 20 minutes. Remove pie weights and parchment paper and continue baking until crust is dry to the touch, about 10 minutes more. Let cool slightly before adding filling, at least 10 minutes.
Meanwhile, bring a medium saucepan of heavily salted water to a boil over high heat. Once boiling, add broccoli and cook until fork tender, about 2 minutes. Drain and let cool in a single layer on a large plate.
Heat olive oil in a medium frying pan over medium-high heat. When it shimmers, add mushroom and onion and season with salt and freshly ground black pepper. Cook until vegetables are soft and mushroom edges are golden, about 5 minutes. Remove from heat and let cool slightly, about 10 minutes.
Whisk together eggs, half-and-half, salt, and pepper in a large bowl until incorporated and smooth. Stir in broccoli, mushroom mixture, and cheese.
Pour custard into pie shell and bake until puffed and golden brown, about 45 minutes. Let cool at least 20 minutes before slicing.
Beverage pairing: J. Moreau & Fils Chablis Premier Cru Vaucoupin, France. French wine for a French-styled dish. The richness of eggs, cream, and Gouda suggests a grape such as Chardonnay, and the mushrooms and broccoli request that it be from France. A medium-bodied wine of moderate richness is called on to balance the equation, making Chablis premier cru a good call. This one has the acidity to cut through the richness, yet the liveliness to work with the vegetables.
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