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For tangzhong, whisk together 1/3 cup milk and 1/3 cup cake flour in a small saucepan. Cook, stirring frequently, over medium heat until a very thick paste forms, about 3 minutes (watch constantly—it thickens quickly). Cool briefly.
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Heat 1 cup of milk in a second saucepan to 105°F to 115°F (40°C to 46°C). Transfer warm milk to blender bowl; Stir in the yeast and leave for 5 minutes.
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Add the evaporated milk, melted butter, eggs, sugar, salt, vanilla, cooled tangzhong, and 2 cups cake flour to a stand mixer. Using the paddle attachment, beat on low speed until the mixture is smooth, about 3 minutes. Gradually whisk in the remaining 2 1/4 cups cake flour to make a soft dough. Continue to beat on low speed until the dough is smooth, 2 to 3 minutes. The dough becomes soft.
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Spread the dough on a floured surface. Knead the dough until smooth, 2 to 3 minutes. Add 2 to 4 tablespoons of additional cake flour. Be careful not to add too much flour. (If the dough starts to stick to the surface, use a scraper to lift the dough off the surface and add a sprinkling of flour. If the dough is too soft to knead, refrigerate covered for 2 hours or up to overnight before kneading.) Form dough into a ball. Transfer to a lightly greased bowl, turning once to greased surface; cover. Leave in a warm place until doubled in size, 1 to 1 1/2 hours.
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Knead and flatten the dough on a floured surface. Roll the dough into a 10-by-15-inch rectangle about 1/2 inch thick. If the dough bounces back, let it rest for an additional 10 minutes. Flour 2 large baking sheets.
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Cut out dough using 3-inch and 1-inch round cutters to make donuts and donut holes. Arrange cutouts 1 inch apart on prepared baking paper. Cover with a towel and let the cutouts rise in a warm place until very light and fluffy, about an hour. (If you have scraps, knead them gently, being careful not to overwork the dough; let it rest for 10 minutes before cutting.)
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Arrange wire racks in two 10″ x 15″ rimmed baking trays (one for cooling and one for brushed donuts).
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About 20 minutes before the donuts are done, pour the oil to a depth of 2 1/2 to 3 inches into a large frying pan or deep-fat fryer. Heat to 365°F (185°C).
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Working in batches of 2 to 3 donuts, fry, once halfway through, until golden, about 90 seconds. Transfer to the prepared cooling rack to cool slightly. Repeat the process with the rest of the donuts. For donut holes, working in batches, fry, stirring constantly with a slotted spoon, until golden, about 1 minute. Transfer to the prepared rack to cool slightly.
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Whisk the powdered sugar, melted butter, vanilla extract, and salt together in a bowl to create a vanilla glaze. Beat in the hot water, 1 tablespoon at a time, until desired consistency. If the glaze has thickened, add additional hot water, 1 teaspoon at a time.
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Dip warm donuts and donut holes in vanilla coating, turning to coat both sides. Arrange cakes on second rack prepared to allow glaze, about 30 minutes.
Chef notes:
Instead of evaporated milk, you can use 1/2 cup plus 2 tablespoons of milk as well.
Store cookies at room temperature in a loosely covered container for up to 24 hours. Freeze unglazed cakes in an airtight container for up to 1 month. Heat the frozen cookies in a 350°F (175°C) oven for 10 minutes. Glaze as directed while still warm.