Make the crust: Pulse the all-purpose flour, granulated sugar and salt in a food processor. Add the butter and pulse until it looks like coarse meal with some pea-size bits. Add the egg and pulse a few times until the dough just starts to clump together. Turn out onto a piece of plastic wrap, gather the dough into a disk and wrap. Refrigerate until firm, at least 2 hours or overnight.
Make the frangipane: Combine the almond flour, granulated sugar, all-purpose flour, butter, egg, brandy (if using), extract, pie spice and salt in a medium bowl and mix with a rubber spatula until smooth. Refrigerate until ready to use.
When ready to assemble the galette, preheat the oven to 400˚ F. Remove the dough from the refrigerator and let sit at room temperature for a few minutes. Roll out the dough on a lightly floured surface into a 12-inch round, about 1/8 inch thick. Transfer to a parchment-lined baking sheet. Spread the frangipane over the dough, leaving a 1 1/2-inch border; refrigerate while preparing the apples.
Prepare the apples: Peel the apples and slice 1/4 inch thick. Toss with the brown sugar, granulated sugar, lemon juice and pie spice in a bowl. Arrange the apples over the frangipane, slightly overlapping. Fold in the edge of the dough, pleating as you go. Brush the edge with the beaten egg and sprinkle with the demerara sugar. Bake until the crust is golden brown and the apples are browned and tender, 35 to 40 minutes. Transfer the baking sheet to a rack and let cool.
Heat the apricot preserves and brandy in a small saucepan over medium heat, whisking, until loose. Strain into a small bowl, pressing on the solids. Brush the glaze over the apples.