Whisk flour, baking soda, salt, ginger, cinnamon, and cloves in a bowl.
Beat butter and brown sugar until smooth. Beat in molasses and egg.
Mix in the dry ingredients just until a soft dough forms. Divide, wrap, and chill for 1 hour.
Heat the oven to 350°F and line baking sheets with parchment.
Roll dough to about 1/4 inch thick on a floured surface. Cut shapes and transfer to the sheets.
Bake 9 to 11 minutes, until the edges are set and the centers still look slightly soft. Cool completely.
Whisk powdered sugar, maple syrup, and milk. Drizzle over the cooled cookies and let the glaze set.