Combine the melted butter, molasses, applesauce, brown sugar, and grated ginger in a small bowl. Whisk until smooth. In a separate bowl or the bowl of a stand mixer, whisk together the flours, baking soda, spices, and salt. Add the wet ingredients to the dry, then mix until the dough comes together. It will be a soft cookie dough. Cover the bowl and chill the dough for at least 2 hours.

When you’re ready to make your cookies, preheat your oven to 350 degrees and line a baking sheet with parchment paper. Remove the chilled dough from the fridge and pot the cane sugar into a shallow bowl. Using a small cookie scoop or tablespoon measure, scoop up portions of the dough, roll them in your hands into a ball, then drop them into the sugar. Roll the ball of dough around until it is covered in sugar. Place the cookies on your prepared baking sheet at least one inch apart. Bake the cookies for 15 minutes. If you prefer a crisp cookie, you can bake your cookies for 18-20 minutes.