Spray a rimmed baking sheet generously with nonstick cooking spray. Set aside.
Add the sugar and ginger to a large saucepan. Massage the ginger into the sugar with your fingers. Turn the heat onto medium-high and stir continuously until the sugar liquefies and no sugar crystals remain. The color will be golden brown. Pour onto the prepared sheet and spread with a spatula until about 1/4-inch thick. Set aside for one hour to harden. Once cool, turn out onto a cutting board and cut into 1-inch and smaller shards. They'll be sharp, so be careful.
Preheat oven to 350 degrees. Line a few large baking sheets with parchment paper. Set aside.
To a stand mixer fitted with a paddle attachment, add butter, granulated sugar and brown sugar and mix on medium speed for 4 minutes until light and fluffy. Turn mixer to low and add eggs, one at a time, making sure each egg is fully blended before adding the next, and scraping down the bowl as needed. Add the molasses and fresh ginger and continue to mix for 1 minute until thoroughly combined. Remove bowl from mixer.
In a medium bowl, whisk together flours, ground ginger, baking soda, cinnamon, cloves and salt. Add this dry mixture to the wet ingredients, stirring until just combined with a few streaks of flour. Add in burnt sugar shards, stirring until evenly distributed.
Place granulated sugar in a small, shallow bowl. Roll two tablespoons of dough into a ball and then roll the ball into the sugar until it's generously coated. Place on baking sheet leaving two inches between the balls.
Bake one sheet at a time in the center of oven for 13 minutes. Cool cookies completely on sheet when done baking.