Mix the butter and sugar together in the bowl of an electric mixer fitted with a paddle attachment until they are just combined. Add the vanilla. In a medium bowl, sift together the flour and salt, then add them to the butter-and-sugar mixture. Mix on low speed until the dough starts to come together. Dump onto a surface dusted with flour and shape into a flat disk. Press the dough into a 10-inch-round or 9-inch-square false-bottom tart pan, making sure that the finished edge is flat. Chill until firm.
Meanwhile, preheat the oven to 350 degrees F.
Butter 1 side of a square of aluminum foil to fit inside the chilled tart and place it, buttered side down, on the pastry. Fill with beans or rice. Bake for 20 minutes. Remove the foil and beans, prick the tart all over with the tines of a fork, and bake again for 20 to 25 minutes more, or until lightly browned. Allow to cool to room temperature.
For the lemon curd:
Remove the zest of the lemons with a vegetable peeler or zester, being careful to avoid the white pith. Squeeze the lemons to make 1/2 cup of juice and set the juice aside. Put the zest in a food processor fitted with a steel blade. Add the sugar and process for 2 to 3 minutes, until the zest is very finely minced. In the bowl of an electric mixer fitted with a paddle attachment, cream the butter with the sugar and lemon zest. Add the eggs, 1 at a time, and then add the lemon juice and salt. Mix until combined.
Pour the mixture into a 2-quart saucepan and cook over low heat, stirring constantly, until thickened, about 10 minutes. The lemon curd will thicken at about 175 degrees F, or just below a simmer. Remove from the heat.
Fill the tart shell with warm lemon curd and allow to set at room temperature.