In a medium bowl, cream together 3/4 cup butter and 1/3 cup confectioners' sugar until smooth. Stir in 1 tablespoon lemon juice and 1 teaspoon lemon zest. Combine the flour and corn starch, and blend into the lemon mixture to form a soft dough. Divide dough in half, roll each half into a log about 8 inches long and 1 inch in diameter. Wrap in plastic wrap and refrigerate for 2 hours, or until firm.
Preheat oven to 350 degrees F (175 degrees C). Cut each roll into 1/4 inch slices and place onto the cookie sheet.
Bake in the preheated oven for 8 to 12 minutes, or until set. Cool completely before frosting.
In a small bowl, blend together 3/4 cup confectioners' sugar, 1/4 cup butter, 1 teaspoon lemon juice, and 1 teaspoon lemon zest with an electric mixer until smooth. Frost cookies and allow to dry before serving or storing.