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Sift together the baking powder, baking soda, and 2½ cups of the flour into a bowl.
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In a bowl, with an electric mixer, beat the yolks and egg white at medium-high speed until fluffy. Slowly add the butter, powdered sugar, brandy, orange zest, and vanilla, beating well after each addition.
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Remove the bowl from the mixer and gradually add the flour mixture by hand, in 1/2-cup increments, mixing in with a spatula. Knead until the dough is elastic and smooth, then divide into 3 balls.
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Lightly flour a work surface. Using a rolling pin, roll out each ball, one at a time, until as thin as a dime. As you roll, rotate the dough and sprinkle lightly with flour, so that it opens up evenly on all sides and becomes elastic and silky. You can also do this using a pasta maker, passing the balls through it in ever-narrowing sequence, until you have long, thin strips.
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There are many shapes for xerotigana, but on Ikaria the preferred shape is the simplest: a strip about 2 inches wide and 4 inches long. Cut the strips and set them aside, covered, until you finish and until you have heated the oil.
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In a large, deep pot, heat 4 inches / 10 cm of oil to 350°F / 175°C. Using tongs or a slotted spoon, place several dough strips at a time in the oil, being careful not to overcrowd them. Fry for a few seconds, until the xerotigana turn light golden and bubble and float to the surface. Remove with a spider or slotted spoon and drain on paper towels. Repeat until all the dough is used up.
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Place the xerotigana on a large platter or platters and drizzle generously with warm honey. Sprinkle with cinnamon and walnuts. Serve.