Beat 1 cup plus 1 1/2 tablespoon butter, confectioners' sugar, white sugar, 1 tablespoon lemon zest, and vanilla extract together in a bowl until smooth. Stir in polenta. Add flour in 2 batches, mixing until dough comes together. Stir in lemon juice.
Turn dough out on a floured work surface; knead gently until smooth, about 1 minute. Divide into 2 pieces. Place on parchment paper; roll up into logs and wrap well.
Refrigerate dough logs until firm, about 1 hour.
Preheat oven to 400 degrees F (200 degrees C). Lightly grease 2 baking sheets.
Cut logs into 3/4-inch slices. Space slices 1 inch apart on baking sheets. Sprinkle remaining 1 tablespoon lemon zest on top.
Bake in the preheated oven until lightly brown, 15 to 20 minutes. Cool on the baking sheets for 5 minutes before transferring to a wire rack to cool completely, about 10 minutes.