Beat together the coconut oil and maple syrup with an electric mixer until fluffy. Add the flax egg and beat until smooth.
Add the dry ingredients and mix until the dough sticks together. Cover and refrigerate for at least 2 hours, preferably overnight.
Preheat the oven to 350 degrees F. Line a baking sheet with parchment paper and set aside.
Line a flat surface with parchment paper, sprinkle with a little flour. Turn the chilled dough onto the surface and allow it to sit for 30 minutes.
Gently knead it with your hands until it's pliable, then roll the dough to ¼” thick and cut out with cookie cutters. Remove the extra dough and transfer the cut cookies onto the baking sheet. Repeat with the rest of the dough until you don’t have any left.
Bake the cookies for 9 – 11 minutes until they’re nicely golden brown.
Cool on a wire rack.
When ready to ice, whisk together the icing ingredients, pour into a piping bag and pipe onto the cookies!