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1. To make the cinnamon sugar, stir together sugar and cinnamon in a small bowl and set aside.
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2. For the doughnuts, combine the flours, yeast, sugar, salt, water, and eggs in the bowl of an electric mixer fitted with the dough hook. Beat on low speed for 2 to 3 minutes, until the dough starts to come together. Beat on low speed for another 3 to 4 minutes, until the dough is stiff and dry.
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3. With the mixer on low speed, add the butter, one piece at a time, mixing after each addition until the butter disappears into the dough. Increase the mixer speed to medium and beat for another 15 minutes, until the dough becomes sticky, soft, and somewhat shiny. Increase the speed to medium-high and beat for another 2 to 3 minutes, until the dough makes a slapping noise against the side of the bowl. The dough is ready when you can gather it into a ball and pick it up in one piece.
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4. Place the dough in a large bowl and cover with plastic wrap, pressing the wrap directly onto the surface of the dough. Let rest in the refrigerator for at least 6 hours, or overnight.
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5. Transfer the dough to a lightly floured work surface and roll out to a 3/4-inch thickness. Using a floured 1-inch cookie cutter, cut the dough into circles. Transfer the doughnuts to a large baking sheet and cover with a clean dish towel. Let rise in a warm, draft-free area for about 1 1/2 hours, until puffed.
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6. Meanwhile, pour enough oil into a large saucepan to come 2 inches up the side of the pan. Heat the oil over medium-high heat, until a deep-fry thermometer registers 350u00b0F. Working in batches, drop the dough into the hot oil and fry for 1 to 2 minutes, turning occasionally, until golden brown. Drain on paper towels, then immediately toss in the cinnamon sugar. Serve immediately.
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The brioche dough in this recipe is from Flour by Joanne Chang.
