Whisk 1/2 cup bread flour and 1/2 cup water in a saucepan until smooth. Cook over medium heat, stirring constantly, until thickened and opaque (5-7 minutes). Let cool completely
In a stand mixer, combine 2 3/4 cups bread flour, 1/2 cup whole milk, 1/2 cup Tangzhong starter, 1 tablespoon yeast, 5 tablespoons softened butter, 3/4 cup sugar, 2 eggs, and 1/2 tsp salt. Knead on medium speed until smooth and elastic (5-7 minutes)
Place dough in a greased bowl, cover, and let rise in the fridge for 24 hours (refrigerated fermentation)
Turn out dough onto a floured surface, divide into 10-12 balls. Roll each ball into a 12-inch circle, cut out a 3-inch center. Fry in oil at 350°F (180°C) until golden brown (2-3 minutes per side)
Whisk 3 cups powdered sugar, 4 tbsp softened butter, and 1 1/2 tsp vanilla until smooth. Adjust with milk if needed. Drizzle over donuts while warm