Special equipment: twenty-four 1 1/2-to-2-inch tart molds
For the crust: Combine the flour, sugar and salt in the bowl of a food processor. Cut the butter into cubes and add to the flour mixture. Pulse until the mixture resembles cornmeal. Add the egg and pulse until the mixture is just wet. Transfer the dough to a clean work surface and knead with the palm of your hand until the pastry just holds together. Do not overwork or the dough will toughen. Wrap in plastic wrap and refrigerate 30 minutes or longer, until the dough is well chilled. Dough may be refrigerated for several days, or frozen for later use.
For the filling: Combine the sugar, water and lemon juice in a saucepan over high heat. Bring to a boil and cook until the sugar is melted and the liquid is reduced by half. Remove from the heat and stir in the coconut. Add the yolks and vanilla, blending well. Set aside to cool.
Dust a clean work surface with flour. Divide the dough in half. One half at a time, roll out the dough as thinly as possible. Cut 12 rounds. Line 12 tartlet molds. Repeat with the remaining dough. Place the molds on a baking sheet and chill at least 15 minutes.
Preheat the oven to 375 degrees F. If available, place an empty tartlet mold on top of each dough-lined mold. Or top with parchment and add pie weights.
Bake until the dough shells are lightly browned, about 20 minutes. While hot, remove the weights and brush the shells with the reserved egg whites. Fill each shell with coconut filling, mounding slightly. Bake until the top of the filling is lightly browned, about 20 minutes.