Line a 12-cup muffin pan with paper liners. Preheat oven to 350°F. Sift flour, ginger, mustard, baking soda, cinnamon, and cloves into a bowl.
Place sugar, molasses, and oil in a large bowl and beat until well combined. Beat in egg whites one at a time until light and well blended. Fold the flour mixture and buttermilk into the sugar mixture alternately, beginning and ending with the flour mixture. Spoon into lined cups. Bake 20 minutes or until a skewer inserted in the center of a cake comes out clean. Place pan on a wire rack and let cupcakes cool completely.
Peel the fresh ginger and grate finely. Squeeze to extract the juice and measure 2 teaspoons. Combine icing sugar and ginger juice in a bowl to make a glaze. Add a little water if necessary. Spread glaze over the tops of the cupcakes and sprinkle with crystallized ginger.