Prepare the tangzhong: In a small saucepan, whisk together the bread flour, milk, and water until smooth. Cook over medium-low heat, stirring constantly, until the mixture thickens into a paste-like consistency (about 3-5 minutes). Remove from heat and let it cool slightly.
Make the dough: In the bowl of a stand mixer fitted with a dough hook, combine the bread flour, yeast, sugar, and salt. Add the cooled tangzhong, eggs, and lukewarm milk. Mix on low speed until a shaggy dough forms. Increase the speed to medium and knead for 8-10 minutes, until the dough is smooth and elastic. Gradually add the softened butter, a tablespoon at a time, allowing each addition to fully incorporate before adding the next. Continue kneading until the dough is smooth, glossy, and pulls away from the sides of the bowl.
First rise: Place the dough in a lightly oiled bowl, turning to coat. Cover with plastic wrap or a damp towel and let it rise in a warm place for 1-1.5 hours, or until doubled in size.
Shape the donuts: Gently punch down the dough to release the air. Turn it out onto a lightly floured surface and roll it out to about ½ inch thickness. Use a donut cutter to cut out donut shapes. Place the cut donuts onto a baking sheet lined with parchment paper. Gather the scraps, gently knead them together, and repeat the rolling and cutting process until all the dough is used.
Second rise: Cover the shaped donuts with plastic wrap or a damp towel and let them rise again for 30-45 minutes, or until they are puffy and almost doubled in size.
Fry the donuts: Heat about 3 inches of vegetable oil in a heavy-bottomed pot or deep fryer to 350°F (175°C). Carefully place a few donuts into the hot oil, being careful not to overcrowd the pot. Fry for 1-2 minutes per side, or until golden brown and cooked through. Remove the donuts with a slotted spoon and place them on a wire rack to drain.
Glaze the donuts: Whisk together the powdered sugar, milk, and vanilla extract until smooth. Dip the warm donuts into the glaze and place them back on the wire rack to allow the glaze to set. Decorate as desired.
