In a bowl, mix together the polenta, flour, baking powder and salt.
In a mixer, cream together the butter and sugar. Add the eggs and yolk one by one, then add the vanilla. Add the dry ingredients and mix until a firm dough forms.
Divide the dough in half, place each piece onto a piece of parchment or wax paper, and shape the dough into a rough log about 2 inches in diameter, dusting with cornmeal as needed to prevent sticking. Roll up in the paper, twisting the ends to seal, and roll back and forth a few times to smooth out the shape. Refrigerate at least 1 hour. (At this point, the dough can be frozen for up to 2 months.)
Preheat the oven to 350 degrees F, and line 2 cookie sheets with parchment paper or silicone baking mats. With a sharp knife, cut the cookies into 1/2-inch slices, and place on the prepared cookie sheets about 2 inches apart. Bake, rotating halfway through, until cookies are just golden, 16 to 18 minutes. Transfer to a rack to cool completely, about 20 minutes.
Polenta cookies can be stored in an airtight container for up to 3 weeks.