Make the crust: Pulse flour, granulated sugar, and 1/2 teaspoon salt in a food processor until combined. Add butter, and pulse until mixture resembles coarse crumbs with some larger pieces remaining, about 10 seconds.
Drizzle 1/4 cup ice water evenly over mixture. Pulse until mixture just begins to hold together (it should not be wet or sticky). If dough is too dry, add more water, 1 tablespoon at a time, and pulse. Press dough into a disk, and wrap in plastic. Refrigerate until firm, about 1 hour or overnight.
Roll dough to a 14-inch circle (1/8 inch thick) on a floured surface. Fit dough into bottom and up sides of a 10-inch fluted round tart pan with a removable bottom. Trim excess dough flush with edges of pan using a knife. Refrigerate until firm, about 1 hour.
Preheat oven to 350 degrees. Prick bottom of tart shell all over with a fork, and fill with pie weights or dried beans. Bake for 30 minutes. Remove weights, and bake until set, about 5 minutes more. Let cool. Leave oven on.
Make the filling: Pulse hazelnuts in a food processor until finely chopped. Add sugars, zest, and 1/4 teaspoon salt; pulse to combine. Add butter, Armagnac, eggs, and vanilla; pulse until mixture is almost smooth.
Spread filling evenly into tart shell. Top with figs and strawberries. Bake for 30 minutes. Reduce oven temperature to 325 degrees; bake until set and dark brown on top, about 1 hour more. Garnish with whipped cream.
Make ahead: Dough can be frozen for up to 1 month. Thaw before using.