In a large bowl, whisk together the oat flour, tapioca starch, baking soda, salt, ginger, cinnamon and coconut sugar.
In a separate large bowl, stir together the wet ingredients (molasses, applesauce, and melted coconut oil) until well combined. Add the dry ingredients to the bowl with the wet and fold together with a spatula until a dough forms. It might look like the dough is dry but don’t worry, just keep stirring. After moist dough has formed, divide it in two separate halves and wrap each one with parchment paper (or plastic wrap). Refrigerate until cold and firm, for at least two hours or overnight.
When you are ready to bake the cookies, take the dough out and preheat the oven to 325°F. Line a hard surface with parchment paper, lightly sprinkle it and a rolling pin with oat flour. Gently roll the dough out on the floured surface to roughly 1/4 to 1/2” thick. Go thinner (1/4") for crispier cookies and thicker (1/2") for softer cookies.
Using a cookie cutter, press shapes into the dough and use your hands to pull away any excess pieces. Form the excess pieces into a large ball and continue to roll out and cut shapes until all of the dough is gone.
Transfer the cookies to two baking sheets lined with parchment paper and bake for 12 minutes. Set aside to cool completely before icing.
Meanwhile, make the icing by combining all of the ingredients (start with 1 tablespoon milk, add a second if needed) in a small bowl and whisking together until smooth. Spoon it into a plastic bag and cut one corner off or use a pastry bag. Slowly pipe the icing on as desired.
Store cookies in an airtight container between layers of parchment paper for up to one week.