Place an oven rack into middle position in oven and preheat oven to 350 degrees F (175 degrees C). Lightly oil an 8x8-inch baking dish.
Place 1 cup flour and butter in a mixing bowl and mash with the back of a spatula or wooden spoon until thoroughly combined. Mix in 1/4 cup confectioners' sugar, vanilla extract, and salt; mash mixture together until mixture looks like a slightly crumbly cookie dough.
Moisten your fingers with a little water and press dough into bottom of prepared baking dish. Use a fork to prick holes all over the crust.
Bake crust on center rack in the preheated oven until crust edges are barely golden brown, 22 minutes.
Beat eggs and egg yolks together in a bowl; whisk in white sugar and 2 tablespoons flour until smooth. Add lemon juice and lemon zest; whisk for 2 minutes. Pour lemon custard over crust.
Bake on center rack until custard is set and top has a thin white sugary crust, 25 minutes. Let cool completely before cutting into bars. Dip knife into very hot water, run around the edge, and cut into 16 squares. Dust cookies with 1 teaspoon confectioners' sugar.