Be sure to freeze your butter and shortening in advance!
In a large bowl, whisk together the gluten-free flour, sugar and salt.
In a small measuring cup, add the egg and vinegar. Whisk until completely smooth, then add cold water until the mixture measures ½ cup and whisk to fully combine. Refrigerate the egg-vinegar mixture until ready to use.
Using the large holes of a box grater, grate the butter. Add the grated butter to the dry ingredients and toss to coat.
Using the large holes of a box grater, grate the shortening. Add the grated shortening to the dry ingredients and toss to coat.
Drizzle 1 tablespoon of the egg-vinegar mixture over the flour and butter mixture. Using a rubber spatula, gently stir and toss to distribute the moisture evenly. Continue adding the egg-vinegar mixture, 1 tablespoon at a time, stirring after each addition, until the dough begins to come together in large clumps. Repeat until you’ve added about 10 tablespoons.
Transfer the dough to a bowl and use your hands to knead it together. You may need to gather and firmly press it a few times until it comes together. If the dough feels too dry, add another tablespoon of the egg-vinegar mixture—but don’t be tempted to add too much liquid! The dough should be moist enough to hold together when pressed, but not sticky.
Turn out the dough onto a lightly floured surface. Gather it together with your hands, pressing any loose bits into the dough to form a cohesive ball.
Gently divide the dough in half and shape each half into a disc about 1 inch thick. Wrap both discs tightly in plastic wrap and refrigerate for 1 hour or up to 2 days.