Making the Dough:
Chop the dried blueberries into 3-4 pieces, then place them in a bowl. Pour about half a cup of boiling water over them and let them sit for 5 minutes, or until the blueberries are soft. Drain the water using a fine mesh sieve.
In a bowl, melt the butter and beat in the sugar until fluffy. Add the sour cream, egg yolks, and vanilla, stirring until well combined.
Mix in the flour, baking powder, and salt until the flour is mostly incorporated. Fold in the blueberries, ensuring the batter is well mixed and there are no flour streaks.
Place the dough on plastic wrap, flatten it to 1-2″ thick, and wrap it tightly. Refrigerate for 1-2 hours or up to 24 hours.
Rolling and Cutting the Donuts:
After chilling, lightly flour your work surface and roll out the dough to about 1 inch thick. Use a 3-4 inch cutter to cut out the donuts, and a 1-inch cutter for the centers.
Gather the dough scraps, re-roll them, and cut more donuts. Limit re-rolling the dough to two times to avoid tough donuts.
Frying the Donuts:
Heat 3-4 inches of a neutral-flavored oil (like canola or vegetable) in a medium pot to 375°F. Use a candy or fry thermometer to monitor the temperature.
Fry 2-3 donuts at a time, cooking the first side for 1 1/2 minutes, then flipping and cooking the second side for another 1 to 1 1/2 minutes until golden brown.
Remove the donuts and place them on a cooling rack set over a baking sheet. Check one donut to ensure it’s cooked through. Adjust the oil temperature as needed.
Glazing the Donuts:
To make the glaze, heat 1/2 cup of milk in a small pot until steaming (but not boiling). Gradually whisk in 4 cups of powdered sugar until the glaze is smooth and flows easily. Add the vanilla extract or bean paste.
Dip each donut into the glaze, coating both sides. Let the excess glaze drip off before placing the donuts back on the cooling rack.