In a mixer fitted with a paddle attachment (or using a hand mixer), cream the butter and confectioners' sugar until smooth. Add the vanilla and mix. With the mixer running at low speed, add the flour and mix. Press the dough into an 8 by 8-inch baking pan. Refrigerate until firm, about 30 minutes. Preheat the oven to 325 degrees F. Prick the dough all over with a fork, then bake until golden brown, about 30 minutes. Let cool.
Cream Cheese Filling: In a mixer fitted with a paddle attachment (or using a hand mixer), cream the cream cheese and sugar together until smooth. Add the eggs and extract and mix at medium speed until smooth. Cover and refrigerate until ready to use.
Lime Filling: In a non-reactive saucepan, whisk the egg yolks, cornstarch, and sugar. Whisk in the water, salt, and lime juice and bring to a simmer over medium-low heat. Whisking constantly, cook until the mixture is thick enough to coat the back of a spoon. Remove from the heat, add the zest and butter, and stir until smooth. Let cool 10 minutes.
Reheat the oven to 325 degrees F. Using a spatula, spread the cream cheese filling evenly over the cooled crust. Spread the lime filling over the cream cheese filling. Bake until the edges are light golden brown, 30 to 40 minutes. Let cool in the pan, on a wire rack, to room temperature. Refrigerate at least 1 hour or overnight before cutting into 16 squares with a sharp knife. Serve chilled.
Recipe courtesy of Gale Gand, Butter Sugar Flour Eggs by Gale Gand, Rick Tramonto, Julia Moskin, Clarkson N. Potter Publishers, 1999