Blend butter and sugar in a large mixing bowl at medium speed with an electric mixer. Beat in egg and extract; set aside.
Combine flour, baking powder and salt; gradually add to butter mixture. Set aside one cup dough. Tint remaining 1 1/3 cups dough with red food coloring and shape into a 3 1/2-inch-long log; wrap in plastic wrap.
Tint 1/3 cup reserved dough with green food coloring; wrap in plastic wrap. Wrap remaining plain dough and refrigerate all 3 doughs 2 hours until firm.
On a lightly floured surface, roll plain dough into an 8"x3 1/2" rectangle. Place red dough log on the end of one short side of rectangle; roll up. Roll green dough into a 10"x3 1/2" rectangle. Place red-and-white dough log on the end of one short side of rectangle; roll up. Wrap in plastic wrap; refrigerate overnight.
Unwrap and cut into 36 (1-inch-thick) slices. Place 2 inches apart on ungreased baking sheets. Place chocolate chips and sesame seed on red part of dough to resemble watermelon seeds. Bake at 350° for 9 to 11 minutes or until firm. Immediately slice cookies in half.