Step 1

In a stand mixer fitted with the paddle attachment, combine the flour, sugar, salt, and butter and mix on the lowest speed until a coarse meal texture forms, about 2 min. With the mixer runing, add the 2 Tbs. water and the vanilla; continue mixing on low until the dough clumps together, about 45 seconds. If the dough remains too dry and crumbly to form a cohesive mass, add a bit more water. Gather the dough into a ball and wrap in plastic, pressing on the plastic to flatten the dough into a disk. Refrigerate until the dough is firm enough to roll, about 30 minutes.

Step 2

On a lightly floured surface, roll the dough to a 1/8-inch thick circle, about 11 inches across. Transfer to a 9-inch deep-dish pie pan and flute the edges, if you like. Freeze the crust until hard, about 40 minutes. Heat the oven to 350°F.

Step 3

When the crust is hard, line it with foil and fill the foil with pie weights or dried beans. Put the pan on a baking sheet, bake for 40 minutes, and then remove the weights and foil and bake until the crust is golden brown and feels dry, another 20 to 30 minutes. Whisk the egg with about 1 tsp. water. Brush the egg on the crust bottom and sides and bake the crust until the egg is set and dry, about 3 minutes.

Step 4

While the pie crust is baking, in a medium bowl, whisk together the eggs and sugar until combined. Add the lime and lemon juices and whisk until smooth. Whisk in the heavy cream and strain the mixture into a pitcher or batter bowl (a large Pyrex measuring cup works well). Stir in the zest. When the crust is done, pour the filling into the crust without removing it from the oven, and reduce the oven temperature to 325°F. Bake the pie until the center is just set, about 50 minutes. Remove the pie from the oven, cool it on a rack, and then refrigerate until cold, at least 6 hours.

Step 5

Put the brown sugar and white sugar in a small, deep, heavy-based saucepan and cover with the water. Put the egg whites and cream of tartar in a stand mixer fitted with the whip attachment. Attach a candy thermometer to the sugar saucepan and bring the sugar to a boil over high heat. When the sugar syrup reaches 248°F, start whipping the egg whites on medium-high speed until they’re very foamy, white, and have increased in volume, about 45 seconds. Keep whipping the egg whites; when the temperature of the syrup hits 254°F, remove the thermometer, set the mixer to medium speed, and carefully and slowly pour about one-third of the sugar syrup into the mixing egg whites (it plops out in drops), avoiding the whip. Add the remaining syrup in a faster, steady stream. Increase the speed to high and whip the whites until they have become voluminous and form firm but not stiff peaks, about 3 minutes; the meringue should still be warm.

Step 6

Scrape the meringue from the bowl onto the chilled pie and, using a rubber spatula, create a smooth dome (avoid pressing on the meringue). With the back of a soup spoon, make decorative peaks in the meringue, working quickly before the meringue cools completely. If you have a kitchen torch, use it to brown the meringue. If not, set a broiler or oven rack to a lower rung and heat the broiler. Set the pie on a baking sheet and put it under the broiler, turning it several times to brown the meringue as evenly as possible.

Step 7

Store the meringue-topped pie in the refrigerator. After a few hours, the meringue will start to weep and break down, so serve the pie as soon as possible.