4 1/2-5 cups bread flour (plus more for kneading, 585-650 g)
Vegetable oil, for frying
Glaze
1 cup powdered sugar (130 g)
1/2 tbsp corn syrup (8 g)
1 tsp vanilla extract (5 g)
1/4 cup whole milk (2 oz)
Directions
Pour the warm milk and water into a stand mixer bowl and sprinkle the yeast over top. (No proofing needed for instant yeast; allow 5 minutes if using active dry yeast until frothy.)
Add the egg, butter, salt, and sugar.
Add 4 1/2 cups flour (reserve the remaining 1/2 cup) and mix with the beater blade until barely combined. Rest 5 minutes for the flour to absorb the liquids.
Remove the beater blade and attach the dough hook.
Beat on medium speed, adding up to 1/2 cup more flour if needed. Knead 5-7 minutes until elastic and smooth; the dough should remain tacky.
Transfer to a greased bowl and cover with a towel or wax paper.
Let rise in a warm place until nearly doubled (30-45 minutes, up to 1 hour maximum).
Turn the dough onto a floured pastry mat and sprinkle with additional flour.
Roll into a 1/2-inch thick rectangle.
Cut donuts with a donut cutter and reroll the scraps. Yields approximately 12-15 donuts.
Place the cut donuts on cookie sheets and let rise 20 minutes while the oil heats.
Heat 2 inches of vegetable oil to 350F in a large pot, Dutch oven, or deep fryer.
Gently lower the donuts into the hot oil (4-5 at a time). Fry the first side about 45 seconds until golden brown, then flip and fry the other side no longer than 1 minute. Remove and cool on a wire rack over a baking sheet or paper towels.
Combine the powdered sugar, corn syrup, vanilla extract, and 1/4 cup whole milk in a small bowl.
Dip the warm donuts into the glaze and return to the cooling rack to set.
Serve warm; store leftovers in an airtight container.
