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.
Dust off excess flour between layers of dough. Press dough layers so they do not stick together. Roll out green dough on parchment paper and use it to support dough when wrapping.