Mix 1/3 cup of the ice water and the sour cream in a small bowl until combined. Process the flour, sugar, and salt together in a food processor until combined. Scatter the butter pieces over the top and pulse the mixture until the butter is the size of large peas, about 10 pulses.
Pour half of the sour cream mixture over the flour mixture and pulse until incorporated, about 3 pulses. Repeat with the remaining sour cream mixture. Pinch the dough with your fingers, if the dough feels dry and does not hold together, sprinkle 1 to 2 tablespoons more ice water over the mixture and pulse until he dough forms large clumps and no dry flour remains, 3 to 5 pulses.
If using for Apple Crumble Pizza Pie, refrigerate dough until it is a consistency suitable for rolling. If using for a pie, divide the dough into 2 even pieces and wrap in plastic wrap, forming into 4-inch disks before refrigerating.