If there's no need to be gluten free, swap out the flours and starches for an equal amount of all-purpose flour. Recipe adapted from Aran Goyoaga and her blog, Cannelle et Vanille.
Make frozen yogurt: In a blender, combine raspberries and superfine sugar and blend until smooth. Pour through a fine-mesh sieve set over a medium bowl, pressing on solids; discard solids. Whisk in yogurt. Freeze in an ice cream maker according to manufacturer's instructions. Transfer mixture to an airtight container and freeze 2 hours (or up to 1 month).
Make cookies: In a medium bowl, whisk together flours, starches, and salt. In a large bowl, using an electric mixer, beat butter and granulated sugar on high until pale and fluffy, about 3 minutes. Add vanilla and egg yolk and beat until combined, scraping down bowl as needed, 1 minute. With mixer on low, gradually add flour mixture and beat until dough comes together (dough will be sticky). Form dough into a disk, wrap tightly in plastic, and refrigerate 2 hours.
Preheat oven to 350 degrees. On a lightly floured work surface, roll out dough to a 1/4-inch-thick round. With a 2 1/2-inch fluted cookie cutter, cut dough into 24 rounds and place on two parchment-lined rimmed baking sheets (reroll scraps if necessary). Freeze 15 minutes, then bake until cookies are lightly golden, 10 to 12 minutes. Let cool completely on sheets on wire racks, 20 minutes. (To store, transfer to an airtight container at room temperature, up to 3 days.) To serve, transfer frozen yogurt to refrigerator to soften, 20 minutes. Scoop 1/4 cup frozen yogurt onto flat sides of 12 cookies and sandwich with remaining cookies.