In a large bowl, cream butter and confectioners' sugar until light and fluffy. Combine flour and cornstarch; gradually add to creamed mixture and mix well. Stir in lime zest and vanilla. Tint half of the dough green if desired.
Using a cookie press fitted with a ribbon disk, press dough into 12-in.-long strips 2-in. apart on ungreased baking sheets. Cut each strip into 3-in. pieces (do not separate pieces).
Bake at 350° for 8-10 minutes or until edges are lightly browned. Cool for 2 minutes before removing to wire racks. Combine glaze ingredients; brush over cooled cookies. Let stand until set.
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