In the bowl of a stand mixer fitted with the paddle attachment, combine sugar, powdered sugar, millet flour, potato starch, tapioca starch, sweet rice flour, salt, xanthan gum, and cardamom. Mix on low speed until evenly combined.
Add the shortening or dairy-free butter. Mix on low speed until the mixture resembles coarse sand and no large chunks of fat remain.
With the mixer on low, slowly drizzle in the ice water. Mix just until the dough begins to come together. The dough should look slightly crumbly but hold together when pressed in your hand. You may not need all of the water.
Turn the dough out onto a piece of parchment paper and gently press it into a flat disk. Let rest for 15 to 20 minutes to allow the flours to hydrate.
Place a second sheet of parchment paper on top and roll the dough to about ⅛-inch thick.
Transfer the rolled dough, still between the parchment sheets, to the refrigerator or freezer. Chill until firm, about 15 to 30 minutes.
Remove the top sheet of parchment and cut out cookies using a round cutter about 1½ to 2 inches in diameter.
Transfer the cut cookies to a parchment-lined baking sheet, spacing them about ½ inch apart. Gather scraps and reroll up to two times, if needed.
Freeze the cut cookie dough rounds for at least 60 to 90 minutes, or until firm.
When ready to bake, preheat the oven to 325°F.
Bake the cookies directly from frozen for 12 to 14 minutes, or until the edges are very lightly golden and the bottoms are just beginning to brown.
Let cool completely on the pan. The cookies will firm up as they cool.