On a lightly floured surface, unroll pie dough sheets. Using a 3 ¼-inch round cutter, cut 12 rounds from dough. Press dough rounds into bottoms and sides of 12 (2 ½-inch) tartlet pans, trimming and discarding excess dough. Using the wide end of a chopstick, press dough into indentations in sides of pans. Place tartlet pans on a rimmed baking sheet, and freeze for 15 minutes. Place a paper cupcake liner or a parchment paper square in each tartlet pan and top with pie weights or dried beans.
Bake until tartlet shells are set, 4 to 6 minutes. Remove weights and paper. Continue baking until lightly browned, approximately 3 minutes more. Let cool completely in pans.
In the work bowl of a food processor, pulse together almonds and sugar just until sandy-fine crumbs result. (Do not overprocess.) Add butter, pulsing until blended. Add egg, flour, and vanilla extract, processing just until all ingredients are incorporated. Divide mixture, approximately 1 tablespoon each, among cooled tartlet shells.
Reduce oven temperature to 375°F.
Bake tartlets until filling is set and golden brown, 10 to 12 minutes. (Filling will puff during baking but will fall when it cools.) Let cool completely on a wire cooling rack before carefully removing tartlets from pans.
Divide raspberries among tartlets.
In a small saucepan, melt apricot preserves over low heat. Press warm preserves through a fine-mesh sieve, discarding solids, to create a syrup. Brush warm syrup over raspberries and tartlet filling. Serve within 4 hours.
Notes
Make-Ahead Tip: Tartlets can be baked up to a day in advance and stored in an airtight container at room temperature. Garnish with raspberries and brush with apricot preserves just before serving.
Recipe by TeaTime Magazine at https://teatimemagazine.com/raspberry-topped-frangipane-tartlets/