Set 2 racks in the middle and upper third of the oven and preheat to 350 degrees F. In a food processor or the bowl of an electric mixer, thoroughly mix together the dry ingredients. On low speed, add the butter, then the corn syrup, milk, and vanilla. Process or beat until the dough is completely blended.
Between sheets of waxed paper, roll the dough to a thickness of 1/8-inch. Refrigerate for 30 minutes, until firm. Cut the dough into animal shapes with cookie cutters or press molds (lightly floured to prevent sticking.) Arrange on nonstick or parchment-lined cookie sheets and refrigerate for 15 minutes to firm the dough. Bake for 7 minutes, until very lightly tanned around the edges. Set the pans on a rack to cool.