Combine the ice water and buttermilk in a measuring pitcher and chill until needed. In a large bowl, combine the sweet rice, oat, and millet flours with the cornstarch, tapioca flour, ground chia seed, sugar, and salt. Scatter the butter pieces over the top, and work in with a pastry blender or your fingertips until the mixture resembles gravel, with lots of butter chunks the size of peas and almonds.
Drizzle the buttermilk mixture into the flour mixture 1 tablespoon at a time, tossing the dough with a flexible silicone spatula to moisten evenly. Add just enough water for the dough to hold together when you give it a squeeze, and add it directly to the dry floury bits that like to hang out on the bottom of the bowl; you may not need all of the mixture, or you may need to add more ice water.
At this point you can do one of two things:
-Knead the dough in the bowl 10-20 times to bring it together.
-Dump the dough out onto the counter and fraisage by dragging portions of the dough across the counter with the heel of your hand (this makes for a flakier dough).
Either way, gather the dough up into a ball (a metal bench scraper helps if using the fraisage method) wrap it loosely in plastic wrap, and flatten it into a disk. Chill the dough 30 minutes.
Optionally, for extra flake, roll the dough out on a lightly floured surface, dusting on top and underneath the dough as you work to prevent sticking, until it is a scant 1/4" thick; the dimensions don't matter. Fold the dough into thirds like you're folding a letter, then fold again into thirds. The dough may crack, and this is ok. Wrap the dough and chill until cold, at least 30 minutes and up to 1 day.
On a lightly floured surface, press and roll the dough into a square, then cut the square in thirds both ways to make 9 squares. Roll each square into a circle 6" in diameter and 1/8" thick. Use a pizza wheel to trim the dough into an even round, and stack the rounds on a plate, dusted with a bit of flour to prevent sticking. Chill until cold, 30 minutes, or wrap and chill up to 1 day.
Position racks in the center and upper third of the oven and preheat to 425ºF. Line two large, rimmed baking sheets with parchment paper.