For the gluten-free flour blend: In a large bowl, whisk together the rice flour, tapioca flour, potato starch, xanthan gum and salt.
For the cookies: In the bowl of a stand mixer fitted with the paddle attachment, beat together the butter, granulated sugar, brown sugar and vanilla on medium-high speed until light and smooth.
Meanwhile, in a small bowl, combine the almond flour and salt with 2 cups gluten-free flour blend. Add to the butter mixture and beat on low speed until just combined. Divide the dough in half and flatten each to make 2 disks; wrap in plastic wrap and chill until slightly firm, at least 20 minutes.
Preheat the oven to 350 degrees F and line 2 baking sheets with parchment paper. Working with one dough disk at a time, on a surface dusted with some gluten-free flour blend, roll out the dough to about 1/4-inch thickness. Using cookie cutters, cut out cookies and place on the prepared baking sheets; refrigerate until firm, about 15 minutes. Bake until the edges are golden, about 10 minutes. Let cool completely on a wire rack. Repeat with the remaining dough.
For the icing: Set a wire rack over a parchment paper-lined baking sheet. In a small bowl, whisk together the confectioners' sugar and whipping cream until smooth. Submerge the cooled cookies into the icing, letting any excess drip off, then place on the prepared wire rack and decorate with sprinkles.