Whisk together flour, baking soda, salt, ginger, cinnamon, cloves, and nutmeg in a bowl and set aside.
Beat butter and brown sugar until fluffy. Mix in molasses and egg until combined. Add the dry ingredients on low speed and mix until a thick, slightly tacky dough forms.
Divide the dough into two flat discs, wrap tightly, and refrigerate for at least 2 hours or up to 24 hours.
Preheat the oven to 350°F. Roll one disc between parchment sheets to 1/4-inch thick. Press cutters straight down and place shapes 1 inch apart on a parchment-lined baking sheet. Refrigerate the filled tray for 10 minutes before baking.
Bake one tray at a time for 8 to 10 minutes, until the edges look set and matte while the centers remain slightly soft. Cool on the tray for 5 minutes, then transfer to a wire rack to cool completely.
For icing, stir powdered sugar with 2 tablespoons water, adding more by the teaspoon until pipeable. Decorate cooled cookies and let icing dry completely before stacking.