In a medium bowl, whisk together the sugar, baking powder, baking soda, salt and cake flour. Create a well in the dry ingredients and add in the egg, coconut milk and coconut oil. Slowly incorporate the wet ingredients into the dry using a wooden spoon or rubber spatula. Mix until dough starts to come together.
Turn dough onto a lightly floured surface and knead a few times until no wet spots remain. Be careful not to over knead, as your donuts will be tough. Lightly roll out the dough into a ¼ inch thick circle. Using a donut cutter or a biscuit cutter about 3 ¾ inch, cut out donut rounds with cutter. If using a biscuit cutter, use a smaller round to cut out donut holes. Combine the scraps and re-roll the dough until you have about 8 donuts.
In a heavy bottomed saucepan, heat up the vegetable oil over medium heat. You can use a thermometer to test when the oil is ready, about 325 degrees. Or you can toss in one of the donut holes in there and when the oil is ready, it will float. Once hot, fry the donuts, in batches, until golden brown, about three minutes each side. Transfer to a wire rack lined with paper towel. Let cool slightly before glazing.
To make the glaze, whisk together all the ingredients until smooth. Dip each donut in glaze and let set for a few minutes.