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.