Roll pate brisee to 1/4-inch thickness on a lightly floured surface. Fit dough into a 10-inch tart pan with a removable bottom; trim dough flush with rim. Prick bottom all over with a fork. Freeze until firm, about 30 minutes.
Preheat oven to 375 degrees. Line dough with parchment, and fill with pie weights or dried beans. Bake until edges turn gold, about 20 minutes. Remove parchment and weights, and bake until crust is golden brown, 15 to 20 minutes. Let cool on a wire rack.
Reduce oven temperature to 350 degrees. Melt butter in a medium saute pan over medium heat until foamy. Add scallions; cook, stirring, for 1 minute. Add morels, and cook, stirring often, until fragrant, for 2 minutes. Stir in milk and cream, and bring to a simmer. Transfer to a medium bowl; let cool to room temperature, about 30 minutes. Stir in salt and pepper.
Place crust on a rimmed baking sheet. Whisk eggs in a medium bowl. Stir in mushroom mixture. Fold in Gruyere. Pour mixture into crust. Bake until puffed and gold, 25 to 30 minutes. Let cool slightly; serve warm or at room temperature.
When choosing morel mushrooms, look for ones that smell slightly earthy or nutty. They should be dry and show no signs of spoilage. An equal amount of fresh porcini or cremini mushrooms can be substituted for the morels. To clean morels, submerge them in cold water and drain on a clean kitchen towel.