In the bowl of a stand mixer fitted with the dough hook, beat together the butter and sugar. Beat in the egg.
In a separate bowl, combine the remaining dry ingredients.
Alternate adding the milk and flour mixture until everything is well combined. Knead for just a couple of minutes. If the dough is too sticky to roll out, add more flour until it's workable.
Dump the dough out on a floured surface and pat flat. Sprinkle dough with half of the blueberries and fold dough over on itself. Flatten and sprinkle with remaining blueberries. Fold dough over to cover blueberries and roll out to 1/4 inch thick. Use a donut cutter to cut donuts.
Add to a pot of 375 degree oil, just a few donuts at a time so as not to crowd the pan, and fry until golden, about 2 minutes each side.
Drain on a paper towel lined plate.
Add powdered sugar, milk, and vanilla to a saucepan over low heat and whisk to combine. Once warmed, dip each donut in the glaze to coat and place on a cooling rack. Let dry 5 minutes before serving.