Preheat the oven to 350ºF (180ºC). Lightly grease a 6-hole donut tin with butter, then fit a piping bag with a large round tip and set aside.

Place the melted butter, sugar, milk, egg, and vanilla in a mixing bowl and whisk until smooth.

Sift the flour into the bowl through a fine mesh sieve. Add the baking powder and salt, then fold gently with a rubber spatula until the flour is about halfway mixed.

Add the blueberries. Continue folding until the blueberries are evenly distributed and no dry bits of flour remain.

Spoon the batter into the prepared piping bag. Pipe a full circle of batter into each donut hole, then divide any remaining batter evenly among them. Smooth the tops lightly with a wet fingertip.

Bake for 13 to 15 minutes. Cool the pan on a wire rack for 10 minutes, then turn the donuts out to cool completely before glazing.

Heat 2 tablespoons of blueberries in a small microwave-safe bowl for 15 to 20 seconds, or until just warm. Mash with a fork, then press them through a fine mesh sieve set over the same bowl to catch the juice. Discard the solids.

Sift the powdered sugar into a mixing bowl. Add the vanilla and ½ teaspoon of blueberry juice. Stir well to combine. Repeat, adding blueberry juice ½ teaspoon at a time, until the icing is thick and ribbony.

Spoon over the cooled donuts and let them set for 20 minutes before serving.