Preheat the oven to 350 degrees F.
In a small bowl, add the currants and orange-flavored liqueur, or water. Soak for 5 minutes, then drain and discard the liquid. Set aside.
Sift together the flour, baking powder, baking soda, and salt in a large bowl. Add the butter and sugar to the bowl of a stand mixer and beat to combine. Gradually add the eggs and vanilla and mix on medium speed until incorporated, (the mixture will look curdled). Scrape down the sides of the bowl. Add the flour mixture all at once and mix again on low speed until a dough forms - scrape again including the beaters. When a dough forms add the currants, zest, and polenta and mix on low until thoroughly combined. Roll into small 1-inch balls, and arrange on parchment lined baking sheets. Bake until golden brown, about 8 to 12 minutes. Remove from the oven and let cool before serving.