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Make the phyllo: Combine the flour, salt, 1 ½ cups water and olive oil in the bowl of a stand mixer outfitted with a dough hook. Knead at medium speed until a dough mass begins to form. Continue kneading for about 8 to 10 minutes, or until the dough is smooth and doesn’t stick to the sides of the bowl. Place in an oiled bowl, cover with plastic wrap and let rest at room temperature for at least 30 minutes, or up to three hours.
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Combine the feta and eggs in a mixing bowl.
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Divide the dough into two equal size balls. Let rest, covered, for a few minutes. On a floured surface roll out the first dough ball to a large circle about 20 inches / 50-cm in diameter. Place a dessert plate in the center of the circle. Using a sharp paring knife and working at equal distances around the plate, cut strips, like radii, emanating from the rim of the plate to the periphery of the dough circle.
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Remove the plate and brush the entire surface of the pastry with melted butter. Next, fold in the first strip like a flap over the center of the pastry circle. Brush the exposed surface with butter. Repeat with each of the strips, folding it over the center like a flap and brushing the exposed surface. Pat around the periphery of the pastry once all the strips have been folded and buttered to form a smooth-edged circle. Repeat with the second ball.
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Cover each circle with plastic wrap and let rest for 20 minutes at room temperature or for up toe 24 hours chilled. If refrigerated, bring it down to room temperature when you are ready to make the pie.
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Preheat oven to 350F / 180C. Brush a 15-inch-round baking pan with butter or olive oil.
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Roll out the first dough ball on floured surface. Cut into 3-inch/ 7 ½-cm-wide strips. Brush each strip with melted butter. Spread about 2 tablespoons down the length of each strip and roll to form a cylinder. Place the first filled, rolled cylinder in the center of the oiled pan, twisting it slightly as you place it in the pan. Repeat with remaining phyllo and filling, placing one end next to the other to continue the spiral, so the the whole pan fills up with coiled, twisted strips. Brush the surface with butter and bake for about 45 minutes to one hour, or until puffed and golden. Remove, cool, and serve.