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For the curd: In a small saucepan, whisk together the sugar and cornstarch. Add the egg yolks and whisk until combined. Whisk in the lemon juice and zest. Bring the mixture just to a simmer over medium, stirring constantly, until thickened, 3 to 5 minutes. Strain the mixture through a fine mesh sieve (use a rubber spatula to help work the curd through). Add the butter to the strained curd and stir to combine. Let the curd cool completely.
For the cookies: Position the oven racks to the upper and lower thirds of the oven and preheat the oven to 350 degrees F. Line two baking sheets with parchment paper.
In a medium bowl, whisk together the flour, baking powder and salt.
In a large bowl, beat the butter and sugar with an electric mixer on medium speed until fluffy, about 3 minutes. Add the egg and lemon zest and beat until well combined. Add the flour mixture and beat until just combined.
Scoop the dough in 1-tablespoon mounds onto the prepared sheets. Roll in between your palms to make perfect rounds, then roll them in sugar and put them back on the sheets at least 1 inch apart. Bake the cookies until just puffed, about 6 minutes. Remove them from the oven and, using a 1/2 teaspoon measuring spoon, make a small well in the center of each cookie. Rotate the sheets and return them to the oven and bake until set and light golden, about 8 minutes more. Transfer the cookies to a rack to cool completely.
Stir the cooled curd to loosen it. Scoop about 1/2 teaspoon of curd into the well in each cookie.
For the meringue: Bring some water to a very low simmer in a small saucepan. In the bowl of a stand mixer, whisk together the sugar and egg whites. Set the egg whites over the saucepan; make sure the bottom of the bowl doesn't touch the water. Whisk the mixture until the sugar is melted, 2 to 3 minutes. Rub a bit between your thumb and forefinger to make sure the sugar is dissolved. Transfer the bowl to the stand mixer fitted with the whisk attachment. Beat the mixture on high until stiff glossy peaks form, about 3 minutes. Add the vanilla and beat to combine.
Arrange all of the filled cookies on an unlined baking sheet.
Preheat the broiler and broil the cookies until the meringue is toasted and golden brown, rotating the baking sheet as needed. Alternatively, you can use a small kitchen torch to lightly brown each of the meringue tops.