In a large mixing bowl, combine the warm milk, sugar, and yeast. Allow it to sit for about 5 minutes until frothy.
Add the egg and olive oil (or coconut oil) to the yeast mixture. Whisk until everything is well combined.
In another bowl, mix together the flour, baking powder, and salt. Gradually add this mixture to the wet ingredients, stirring until a soft dough forms.
Turn the dough out onto a floured surface and knead it for about 5 minutes until it becomes smooth and elastic.
Place the dough in a greased bowl, cover it with a clean kitchen towel, and let it rise in a warm place for about 1 hour, or until it has doubled in size.
Once the dough has risen, punch it down and roll it out to about 1/2 inch thick. Use a donut cutter to cut out donuts and holes.
In a frying pan or deep fryer, heat oil to 350°F (175°C). Carefully place the donuts in the hot oil, frying a few at a time.
Fry for about 1-2 minutes on each side until golden brown. Remove them with a slotted spoon and let them drain on a cooling rack.
While the donuts are cooling, mix together the confectioners sugar and 2 tablespoons of milk in a bowl until smooth.
Dip the warm donuts into the glaze, ensuring they're coated well. Allow the excess glaze to drip off, then place the glazed donuts back on the cooling rack.