Make the tangzhong
In a saucepan, whisk water and flour over medium heat for 4–5 minutes until thickened into a paste. Transfer to a bowl and let cool.
Prepare the dough
In a stand mixer, combine flour, sugar, yeast, and salt. Add milk, egg, and tangzhong. Mix on low for 2 minutes until dough forms.
Incorporate butter
Add softened butter gradually, one tablespoon at a time. Once incorporated, knead for about 12 minutes until smooth and elastic (windowpane stage).
First rise
Shape into a ball and place in a greased bowl. Cover and let rise for 30–45 minutes.
Shape the donuts
Roll dough to about 3/4-inch thickness. Cut out donuts and place each on parchment squares. Re-roll scraps as needed.
Second rise
Cover loosely and let rise for 45–65 minutes until puffy and doubled.
Heat oil
Heat 3–4 inches of oil in a heavy pot to 350°F (180°C).
Fry donuts
Fry 2–3 donuts at a time for about 2½ minutes total, flipping halfway, until golden brown. Remove and drain on a wire rack.
Make the glaze
Whisk powdered sugar, hot water, vanilla, and salt until smooth and pourable.
Glaze the donuts
Dip donuts into glaze, allow excess to drip off, then place back on rack.
Set and serve
Let glaze set for 45–60 minutes, then enjoy.