For COOKIES: Combine corn starch, flour and 2/3 cup powdered sugar in medium bowl. Set aside.
Beat 2 cups butter in a large bowl with a mixer at medium speed until softened and smooth. Add the corn starch mixture, vanilla and 1-1/2 teaspoons lemon peel; beat well. (If necessary, refrigerate dough 1 hour or until easy to handle.)
Shape the dough into 1-inch balls and place them 1-1/2 inches apart on ungreased cookie sheets. Flatten slightly with bottom of a glass dipped in sugar to prevent sticking. Bake 10 to 15 minutes, or until edges are lightly browned. Cool completely on wire racks.
For ICING: Beat together in a medium bowl 1/3 cup butter and 1 teaspoon lemon peel until butter is softened. Add 4 cups powdered sugar and lemon juice; beat until well combined, adding additional tablespoons lemon juice or milk, if needed, to reach spreadable consistency. Add red food coloring, if desired. Decorate cookies with icing. Store in a tightly covered container.