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Preheat oven to 350 degrees. Make the crust: Stir together cookie crumbs and melted butter. Press into bottom and up sides of a 9 1/2-inch fluted tart pan with a removable bottom. Refrigerate until chilled, about 30 minutes.
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Bake until firm and darkened, 10 to 12 minutes. Let cool completely on a baking sheet. Meanwhile, make the filling: Heat cream in a saucepan until just starting to boil. Pour over chocolate in a bowl. Let stand for 1 minute, then stir. Whisk in egg, milk, and butter. Pour into cooled crust.
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Reduce oven temperature to 300 degrees. Bake tart until edges of filling are set but center still jiggles slightly, 20 to 24 minutes. Transfer to a wire rack. Serve warm or at room temperature. Top with creme fraiche.
Cook’s Notes
Tart can be refrigerated for up to 1 day.