Position racks in the middle and lower third of the oven; preheat to 350 degrees F. Make the dough: Pulse the flour, granulated sugar and salt in a food processor until combined. Add the butter; pulse until the mixture looks like coarse meal with pea-size bits of butter. Beat the egg with 1 tablespoon ice water in a bowl, then add to the processor and pulse once or twice. (Stop before the dough gathers into a ball.) Turn out onto a sheet of plastic wrap and pat into a disk. Wrap tightly and refrigerate until firm, at least 1 hour.
Meanwhile, prepare the filling: Toast the walnuts on a baking sheet on the middle oven rack until golden, 7 to 8 minutes; let cool. Process the nuts, 1/4 cup granulated sugar, the egg, 1 tablespoon butter, the vanilla and salt in a clean food processor to make a creamy paste.
Line a flat baking sheet (or an upside-down rimmed one) with parchment paper. Roll the dough into a 12-inch round on a floured surface. Roll the dough up onto the rolling pin, then unroll onto the parchment. Spread the walnut filling over the dough, leaving a 1 1/2-inch border around the edges. Chill while you prepare the apples.
Increase the oven temperature to 400 degrees F. Peel, core and halve the apples, then cut each half into 8 wedges. Arrange over the filling in a circular pattern, slightly overlapping. Fold the edges of the dough inward, pleating it. Sprinkle the apples with the remaining 1 tablespoon granulated sugar and 2 tablespoons butter.
Bake the galette on the middle oven rack until golden, 40 to 45 minutes. (Put a baking sheet on the lower rack to catch any drips.) Cool on a rack. Whisk 1 tablespoon water and the preserves in a bowl. Strain, then brush over the apples. Dust with confectioners' sugar, if desired.
Photographs by Con Poulos
Recipe courtesy Food Network Magazine