Make the brown butter: Melt butter in a medium saucepan over medium heat. Cook, swirling occasionally, until golden brown and nutty smelling, 9 to 10 minutes. Transfer ⅔ of the butter (about 6 ounces) to a bowl and ⅓ (about 3 ounces) to a separate bowl (scraping the brown bits from the bottom of the pan with a spatula). Cover and refrigerate until firm and cold, at least 2 hours and up to 1 week.

Make the crust: Chop the ⅔ portion of brown butter into pieces. Whisk together flour, salt, and granulated sugar in a bowl. Cut butter into flour using two forks or a pastry blender until it resembles coarse meal with several pea-size pieces remaining. Add ice water, 1 tablespoon at a time, using a fork to pull dough together into a crumbly pile (add up to an additional 2 tablespoons of water if needed). Divide into two piles; wrap each in plastic wrap and flatten into discs. Refrigerate until firm, at least 2 hours or up to overnight.

Make the filling: Cook apples and remaining ⅓ portion of brown butter in a large skillet over medium heat, stirring occasionally, until crisp-tender, 10 to 15 minutes. Add brown sugar, flour, lemon juice, salt, cinnamon, nutmeg, cloves, and cardamom. Cook, stirring, until combined and thickened, 2 to 3 minutes. Remove from heat and stir in vanilla. Transfer to a rimmed baking sheet to cool, about 30 minutes.

Preheat oven to 425°F with rack in the bottom position. Line a rimmed baking sheet with aluminum foil. On a lightly floured work surface, roll one disc of dough to a 12-inch circle. Fit into the bottom and up the sides of a 9-inch pie plate. Top with apple filling. On a lightly floured work surface, roll remaining dough to a 12-inch circle. Place on top of filling. Trim both layers of dough to a 1-inch overhang, fold under, and crimp. Freeze 20 minutes.

Whisk together egg and 1 teaspoon water in a bowl. Brush on pie, then sprinkle with turbinado sugar. Cut 3 or 4 slits in pie. Bake on prepared baking sheet 15 minutes. Reduce oven to 350°F, and bake until crust is golden brown and filling is bubbly, 45 to 50 minutes. Cool on a wire rack at least 4 hours before serving.