Warm the milk slightly, then mix in the sugar and yeast. Let it stand for 10 minutes, or until foamy.
In a large bowl, sift the flour, then add the salt, vanilla extract, egg, and melted butter.
Pour in the yeast mixture and knead the dough for 8-10 minutes, until smooth and elastic.
Cover the dough with a clean kitchen towel and let it rise in a warm place for about 1 hour, or until doubled in size.
On a lightly floured surface, roll out the dough to about 2 cm thickness, then use a glass or doughnut cutter to cut out circles.
Place a spoonful of adzuki bean paste in the center of each dough circle, then fold over and press the edges firmly to seal.
Heat the oil in a deep skillet or fryer to about 340°F (170°C).
Fry the doughnuts for 2-3 minutes per side, until golden brown. Remove with a slotted spoon and place on paper towels to drain excess oil.
While still warm, sprinkle with powdered sugar and serve.
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