Be sure to freeze your butter and shortening in advance!
In a medium 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 in the flour mixture.
Using the large holes of a box grater, grate the shortening. Add the grated shortening to the dry ingredients and toss to coat in the flour mixture.
Drizzle 1 tablespoon of the egg-vinegar-water mixture over the flour and butter mixture. Using a rubber spatula, gently stir and toss the mixture, distributing the moisture evenly. Continue adding the liquid, 1 tablespoon at a time, stirring after each addition, until the dough begins to come together in large clumps. Repeat until you’ve added 5-6 tablespoons of the liquid. (Note that you will only use about ½ of the liquid.)
Pour the dough into a bowl and use your hands to gently bring the dough together, being careful not to overwork it. Add another tablespoon of liquid if needed. Don’t be tempted to add too much liquid! It should be moist enough to hold together when pressed but not sticky.
Turn the dough out onto a lightly floured surface. Gather it together with your hands, pressing any loose bits into the dough to form a cohesive ball.
Gently shape it into a disc about 1 inch thick. Wrap the disc tightly in plastic wrap and refrigerate for 1 hour or up to 2 days.