• In a small saucepan, whisk together the water and bread flour. Cook over medium heat, whisking constantly until the mixture thickens to the same consistency as pudding (about 4-5 minutes). Scrape into a small bowl and set it aside until ready to use.
• In the bowl of a stand mixer fitted with the dough hook, combine the bread flour, sugar, yeast, and salt. Add the milk, egg and tangzhong. Knead on low speed until a dough forms.
• With the mixer running on low, add the softened butter a tablespoon at a time, letting it fully incorporate before adding more. Once all the butter is added, increase the speed to a 2 on a Kitchen Aid and knead for 12 minutes, or until the dough is smooth, elastic, and passed the windowpane test.
• Shape the dough into a ball and place it in a lightly greased bowl. Cover and let it rest at room temperature about 30-45 minutes.
• Meanwhile, cut out 8-9 squares of parchment paper.
• Turn the dough onto a floured surface, and lightly flour the top and roll to 3/4 inch thickness. Use a large round donut cutter to cut donuts (you will get about 8 donuts rolled to 3/4 inch). I used a square biscuit cutter for the maple glazed donuts.
• Gently knead the scraps together, then let the dough rest for 5-10 minutes before rolling out and cutting more.
• Place each donut on a square of parchment paper and arrange them on a baking tray. Cover the top loosely with plastic wrap and let rise until puffy and doubled, about 45-65 minutes.
• When the donuts are about halfway through their final proof, start heating 3-4 inches of oil in a heavy pot to 350. Line a baking sheet with a wire rack. Keep an eye on the temperature. If the oil reaches 350 before the donuts are ready, lower the heat slightly to hold it steady until they finish proofing.
• Working in small batches, use a metal slotted spoon to carefully lower 2-3 donuts (with the parchment) into the hot oil, then peel away the paper with tongs. Fry for about 2 1/2 minutes total, flipping once halfway through, until golden brown. Then, transfer to the wire rack to drain.

Make the vanilla glaze

• In a medium bowl, whisk together the powdered sugar, hot water, vanilla, and salt until smooth. Adjust with more water or powdered sugar until you have a thick, yet runny glaze.
• Dip both sides of the donuts in the glaze, let any excess drip off, then place them back on the wire rack over the baking sheet.
• The donuts will need roughly 45-60 minutes for the glaze to fully set, then enjoy!

Maple frosting

• Mix all ingredients together in a large bowl and beat well with a mixture until the ingredients become light and fluffy. Add more milk if needed.
• Wait until the donuts have slightly cooled and then frost generously with the maple frosting.