1. In a medium bowl, whisk together flour, baking soda, ginger, cinnamon, cloves, and salt. In a large bowl, using an electric mixer, beat brown sugar and butter on medium-high until light and fluffy, 3 minutes. Add egg and molasses and beat to combine. With mixer on low, gradually add flour mixture and beat to combine. Form dough into 2 disks, wrap in plastic, and refrigerate 30 minutes.

2. Preheat oven to 350 degrees, with racks in upper and lower thirds. Working with 1 disk at a time, roll out dough between floured parchment paper to 1/8-inch thickness. Stack dough in parchment on a baking sheet; refrigerate until firm, 30 minutes. With a 3 1/2-inch bell-shaped cutter, cut out dough (reroll scraps, if desired). Place cookies, 1 inch apart, on two parchment-lined baking sheets.

3. Bake until cookies are firm and edges begin to brown, 14 minutes, rotating sheets halfway through. Transfer cookies to wire racks and let cool completely.

4. In a large bowl, using mixer, beat confectioners' sugar, egg whites, and cream of tartar on low until combined. With mixer on high, beat until icing is whipped and opaque, 4 minutes. Thin with water as needed to make a stiff but spreadable icing. Divide icing among bowls; tint with food coloring. Transfer icings to large separate zip-top bags and snip 1/8-inch hole in one corner of each bag. Frost cookies in one color, then pipe horizontal lines in a different color; drag a toothpick up and down through lines. Let sit until icing is set, 2 hours.