Preheat a doughnut maker. If using a doughnut pan, preheat the oven to 350°F and grease the pan liberally with palm shortening.

In a blender, combine the eggs, coconut milk, oil, maple syrup, vinegar, vanilla, cocoa powder, coconut sugar, coconut flour, arrowroot, baking soda, and salt and blend on high speed for 30 seconds, until well incorporated. Let the batter rest for 2 minutes, then blend again on high for 15 seconds. Scoop the batter into a large resealable plastic bag, seal the top, and snip one of the bottom corners to make a piping bag. Pipe the batter into the doughnut mold or pan, filling it completely. Cook until the doughnut machine indicator light goes off, about 5 minutes, or bake for 15-17 minutes in the oven, until a toothpick inserted into the center of the doughnut comes out clean. Remove the doughnuts from the mold or pan and cool on a wire rack.

To make the glaze, bring 1 inch of water in a saucepan to a simmer. Place the chocolate, maple syrup, and coconut oil in a heatproof bowl and set it over the simmering water, creating a double boiler. Turn the heat to medium-low and stir the mixture frequently until the chocolate melts and the mixture is smooth. Turn off the heat, remove the bowl from the pan, and stir in 1 tablespoon water.

Pour the sprinkles into a shallow dish. Dip the tops of the cooled doughnuts in the warm glaze and twist gently as you remove them from the bowl to allow excess glaze to drip off. Place the doughnuts on a wire rack to set for 10 minutes, then dip the tops of the doughnuts a second time. Immediately dip the tops in the sprinkles. Serve immediately, or chill in the refrigerator for 1 hour to harden the chocolate.

Store the doughnuts in an airtight container in the refrigerator for 2 weeks, or freeze the doughnuts in an airtight container, with a square of parchment paper placed between each row of doughnuts, for 4 months. Defrost at room temperature for 1 hour before serving.