Shaker-Style Lemon Bars

In a bowl, using an electric mixer, beat the 1 stick plus 2 tablespoons of butter with the sugar and egg yolk until smooth. Add the flour, baking powder, and salt and beat at low speed until evenly moistened, about 2 minutes. Using your hands, knead once or twice to bring the dough together. Using a mandoline or very sharp knife, slice the lemon halves as thinly as possible and transfer to a medium bowl; pick out the seeds. Stir in the lemon juice and sugar. Cover with plastic wrap and refrigerate overnight. Butter a 9-inch square metal baking pan. Pat three-fourths of the dough into the baking pan in an even layer. Roll the remaining dough into eight 9-inch-long ropes. Transfer to a plate. Refrigerate the crust and ropes until firm, about 10 minutes. Using an electric mixer, beat the flour, butter, and eggs until smooth. Fold in the lemon-sugar mixture. Preheat the oven to 350 degrees F. Spread the filling over the dough and arrange the ropes of dough on top in a crisscross pattern. Bake in the lower third of the oven for 40 to 45 minutes, or until the ropes are golden. Let cool completely. Run the tip of a knife around the edge of the pan. Cut the square into 18 bars and serve.