Mix 1 cup of water, 1 cup all-purpose flour, 1 cup bread flour, and the sourdough starter in a large bowl until well combined. Form into a ball, cover, and let it rest for 1 hour.
In a small saucepan, warm the milk, butter, and sugar over medium heat until the butter melts and sugar dissolves. Cool to 85°F.
Transfer the dough to a stand mixer equipped with a dough hook. Slowly pour in the milk mixture and knead.
Add eggs one at a time, followed by the remaining flours. Knead for 8 minutes, then hand-knead as needed until the dough is smooth and only slightly sticky.
Transfer to a greased bowl; let rest for 30 minutes.
Add salt and 2 tablespoons of water. Knead again for 10 minutes until soft. Rest for 5 hours or refrigerate overnight.
Roll the dough to a 1/2-inch thickness, cut doughnuts, and proof for an additional 3 hours (or refrigerate overnight).
Fry at 360°F until golden, about 1-2 minutes per side. Cool and coat with glaze or cinnamon sugar.