In a large bowl, whisk together the flours, sugar, and salt. Add the butter and toss to coat it in the flour mixture. Using your hands, smash the butter between your palms and fingertips, mixing it into the flour, creating long, thin, flaky, floury, buttery bits. Once most of the butter is incorporated and there are no large chunks remaining, dump the flour mixture onto a work surface.
In a measuring cup, combine the vinegar and ice water and drizzle it over the flour/butter mixture. Run your fingers through the mixture like you’re running your fingers through your hair, just to evenly distribute the water through the flour until the dough starts coming together.
Knead the dough a few more times, just to gather up any dry bits from the bottom and place them on the top to be incorporated. You will be tempted to add a tablespoon or two more water now—resist if you can, add only if you must.
Once you’ve got a shaggy mass of dough (it will not be smooth and it certainly will not be shiny), knead it once or twice more and divide it in half. Pat each piece into a flat disc about 1 inch thick. Wrap each disc individually in plastic wrap and refrigerate at least 2 hours.