1. Preheat oven to 350°.
2. To prepare cookies, weigh or lightly spoon flour into dry measuring cups; level with a knife. Combine flour, ginger, baking soda, cinnamon, and salt, stirring mixture well with a whisk.
3. Place brown sugar and butter in a bowl. Beat with a mixer at medium speed 2 minutes or until light and fluffy. Add cane syrup, molasses, and egg; beat 1 minute or until well combined. Add flour mixture, and beat on low speed 1 minute or until just combined. Gently press mixture into a disc, and wrap in plastic wrap. Chill for 30 minutes.
4. Divide dough in half. Roll each dough portion to a 1/8-inch thickness on a lightly floured surface; cut with 5-inch cutters to form 26 cookies. (Reroll scraps, as necessary.) Place cookies 1 inch apart on a baking sheet covered with parchment paper. Bake at 350° for 8 minutes or until lightly browned. Let cool on baking sheet 5 minutes. Remove cookies from baking sheet; cool completely on a wire rack.
5. To prepare icing, combine powdered sugar and milk, stirring with a whisk until smooth. Spoon icing into a small zip-top plastic bag. Snip a tiny hole off one corner of the bag. Pipe icing onto cookies as desired.