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1. Combine flour and cornmeal in a medium bowl. Beat the butter, 3/4 cup sugar, and salt in a large mixing bowl at medium speed until light and fluffy. Add the egg yolks and vanilla and beat until well blended. Beat in flour mixture at low speed just until combined. Beat in currants. Let dough stand 10 minutes.
2. Heat oven to 350 degree F. Roll dough into 1-inch balls. Pour remaining sugar in a shallow bowl. Roll each dough ball lightly in sugar, and place 3 inches apart on a lightly greased cookie sheet. Flatten each cookie to 1/4-inch thick with the bottom of a drinking glass that has been dipped in sugar. Bake until edges are golden brown, 10 to 12 minutes. Transfer cookies to a wire rack to cool. Repeat with remaining dough. Makes about 4 dozen.