In a large bowl, beat sugar and butter until creamy. Beat in egg and extracts until blended. Add flour and beat until well combined. Transfer half of dough to a small bowl. Knead in about 15 drops of red food coloring. Transfer other half of dough to another bowl and knead in green food coloring.
On a sheet of waxed paper, shape red dough into a 61/2-x-2-x-1-inch block. Wrap in plastic wrap and set aside. Repeat with green dough. Refrigerate both blocks for two hours or place in freezer for 30 minutes.
Cut each block of dough lengthwise in half to make two 61/2-x-1-x-1-inch blocks. Brush one side of one red strip with milk. Place brushed side next to green strip. Repeat with remaining two strips. Brush top of one of the red-green blocks with milk. Place second red-green block on top, reversing colors so ends form checkerboards. Wrap in plastic wrap. Refrigerate four hours or overnight.
Preheat oven to 375°. Grease two large cookie sheets. Remove checkerboard block from refrigerator and place on cutting board. Cut dough crosswise into 1/4-inch slices. Place slices one inch apart on cookie sheets. Bake cookies on two oven racks for eight to 10 minutes, rotating cookie sheets between upper and lower racks halfway through baking. Leave cookies on sheets for one minute to cool slightly. Repeat with remaining dough.