Step 1To make the dough using a food processor, pulse together the flour, salt and sugar until thoroughly combined. Add the shortening and pulse until incorporated (the dough will look like moist sand). Add the butter and pulse just until the butter is reduced to small, pea-sized pieces. Sprinkle the vinegar and water over the mixture, and pulse once or twice until incorporated. Remove the crumbly mixture to a large bowl and gently press the mixture together until it comes together to form a dough. Mold the dough into a disk roughly 6 inches in diameter. Cover the disk tightly with plastic wrap and refrigerate at least 2 hours.
Step 2To make the dough by hand, whisk together the flour, salt and sugar in a large bowl. Add the shortening and incorporate using a pastry cutter or fork (the dough will look like moist sand). Cut in the butter just until it is reduced to small, pea-sized pieces. Sprinkle the vinegar and water over the mixture, and stir together just until incorporated. Proceed as above.
Step 3Form the mini-pies: Prepare the muffin tins by greasing every other cup (fill alternating muffin cups to allow extra room to form and crimp the mini-crusts and to allow the heat to circulate more evenly to bake the crust as it bakes). On a lightly floured surface, roll out the dough into a round one-eighth-inch thick and roughly 14 inches in diameter. Cut the dough into rounds using a 5 1/2-inch cutter. Gather the scraps and re-roll, cutting more large rounds (for bottom crusts) or smaller rounds (about 3 inches) to make the top crust. Carefully place the rounds in the muffin cups, pressing the dough against the bottom and edges of the cups. The dough should extend above the cups by one-half to three-fourths inch; roll the edges and crimp as desired.
Step 4If pre-baking the mini-pie crusts, gently line the shells with foil and fill with pie weights. Bake in a 425-degree oven for 10 minutes, then remove the weights and foil. Brush the edges of the mini crusts with egg wash and prick the bottom once or twice with a fork. Continue baking until the bottom of the crust is dry and lightly colored, 6 to 8 minutes more.