Put the flour, sugar, and salt in a food processor. Pulse 3 to 4 times to blend. Distribute the butter in the bowl and pulse 7 to 8 times. Process until the mixture resembles coarse meal, 8 to 10 seconds. In a small bowl, beat the egg, egg yolk, water, and vanilla with a fork. Pour the egg mixture over the flour mixture and pulse 5 to 6 times. Process until the mixture just begins to form a mass, 8 to 10 seconds. Empty the dough onto a lightly floured surface and knead 6 to 8 times until the dough is just smooth and malleable. Shape it into an evenly thick 6-inch square. Using a pastry scraper or the dull side of a long knife, score the dough at 1-inch intervals so you get thirty-six 1-inch squares. Cover the dough with plastic wrap and chill for at least 20 minutes.

You can freeze the dough, still wrapped in plastic wrap, for up to one month. Let it fully thaw in the refrigerator before using in the tartlets.