In a large bowl, cream butter and sugar until light and fluffy. Beat in egg and extract. Combine the flour, salt and baking powder; gradually add to creamed mixture and mix well.
Remove 1 cup of dough; set aside. Add enough red food coloring to tint remaining dough deep red. Roll into a 3-1/2-in.-long log; wrap in plastic and refrigerate until firm, about 2 hours.
Divide the 1 cup of reserved dough into two pieces. To one piece, add enough green food coloring to tint dough deep green. Leave remaining dough plain. Wrap each piece separately in plastic; chill until firm, about 1 hour.
On a floured sheet of waxed paper, roll white dough into a 8-1/2x3-1/2-in. rectangle. Place red dough along short end of rectangle. Roll up and encircle red dough with white dough; set aside.
On floured waxed paper, roll the green dough into a 10x3-1/2-in. rectangle. Place log of red/white dough along the short end of green dough. Roll up and encircle log with green dough. Cover tightly with plastic wrap; refrigerate at least 8 hours or overnight.
Unwrap dough and cut into 1/8-in. slices. Place 1 in. apart on ungreased baking sheets. Lightly press miniature chocolate chips into each slice to resemble watermelon seeds; if desired, add sesame seeds.
Bake at 375° for 6-8 minutes or until cookies are firm but not brown. While still warm, cut each cookie in half or into pie-shaped wedges. Remove to wire racks to cool.