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Whisk together the flour, sugar, and salt in a bowl. Use a spoon to create a well in the middle.
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Add yeast, pour the lukewarm water over the yeast. Wait until it bubbles.
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Add the and oil and knead everything to a smooth dough.
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Cover the bowl with a dishcloth and stow in a warm place for 45 minutes.
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Preheat oven to 350 degrees Line a baking sheet with parchment paper.
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Knead the dough briefly. Divide into three equal portions (use a kitchen scale if you have one.)
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Roll out the dough pieces on a floured work surface into 12-inch rounds. Trim the edges with a sharp knife if they look ragged.
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Spread the one circle with red pesto, then sprinkle half of the cheese over the top.
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Lay the second circle on top, gently spread the green pesto on the second circle and sprinkle the rest of the cheese on top.
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Lay the third circle on top and transfer to the prepared baking sheet.
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Press a 2+1/2 inches in diameter cookie cutter or glass into the center to create an impression. Use a sharp knife to cut a circle out of the impression.
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Starting at the edges, use a pizza cutter or a sharp knife to cut 12 equal-sized strips ending at the edge of the circle cutout.
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Twist each strip once all in the same direction.
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Lightly brush water over everything.
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Place in the oven and bake in the oven for 30 minutes.
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Remove from oven and serve warm.