Pour flour, sugar, and salt together in a food processor and pulse to combine.
Sprinkle shortening chunks over the flour mixture. Pulse in the food processor until mixture has a sandy texture, 3 to 5 one-second pulses.
Scatter butter pieces over the top of the flour mixture. Pulse in the food processor until butter is the size of small peas, 10 to 15 one-second pulses.
Sprinkle water over the top of the butter-flour mixture. Pulse until mixture just starts to come together, adding water 1 tablespoon at a time if necessary. Turn dough out into a bowl. Gently squeeze dough together with your hands to form a ball; the dough should hold together but also still be slightly crumbly. Divide dough into two balls and wrap with plastic wrap. Use a rolling pin to flatten the ball into a disc. Refrigerate until well chilled, at least 2 hours.