Preheat the oven to 350 degrees F (175 degrees C). Line 2 baking sheets with parchment paper.
Combine sugar, brown sugar, and butter in a bowl and beat using an electric blender until fluffy. Crack the egg into a cup, add maple syrup, and lightly beat with a fork. Add to sugar-butter mixture and beat to combine.
Sift flour, allspice, and cinnamon together in a bowl. Gradually add to sugar-butter mixture, 2 tablespoons at a time, beating dough well between each addition until all ingredients are combined.
Take about 1 tablespoon of dough, roll it into a ball using your hands, and place onto the prepared baking sheet. Press down lightly with the palm of your hand to flatten slightly Repeat with remaining dough, leaving space between cookies.
Bake in the preheated oven until slightly golden, 8 to 10 minutes. They may look slightly undercooked when done but don't leave them in too long, so you don't end up with a hard cookie. Transfer to a wire rack to cool completely, about 30 minutes.
Sift confectioners' sugar into a clean, dry bowl. Add maple syrup and mix together with a metal spoon. Thin out icing with a couple of drops of warm water until desired thickness is reached. Drizzle or spread over cooled cookies.