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In a bowl, combine flour and 1/2 cup of the water, and apple cider vinegar. Stir gently to mix and cover with a sealed lid or plastic wrap so the bowl is air tight. Allow to rest for at least 6 hours.
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Place the flour, water and vinegar mixture in a mixer bowl, add the additional 1/4 cup water, yeast, olive oil and salt and knead with dough hook on medium for 2 to 3 minutes. Alternatively, you can knead the dough in a mixing bowl with a wooden spoon or silicone spatula for about 5 minutes by hand until smooth and elastic.
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Cover the bowl and place in a warm (oven light on, no heat) oven for 1 hour, or until doubled in size.
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Twenty minutes before the dough is finished rising, prep your vegetables and greens, so they are ready to be set out on the table.
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After the dough has risen, remove it from the oven.
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Place a baking tile or stone in the bottom of the oven, and remove the top rack so you can easily move the pitas in and out from the baking stone as you bake. Preheat the oven to 475°F.
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Remove the dough from the bowl with a spatula onto a floured board and form it into a rectangle. Cut into six equal squares, placing the squares on a separate board or on a piece of parchment paper on the counter.
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Allow the squares to rest for 5 to 10 minutes while the oven preheats.
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Roll each square into a rough circle of 6 to 8 inches, rolling out 1 or 2 at a time. Open the oven and gently toss one pita dough onto the baking stone. Close the oven and set the timer for 2 minutes. Roll out the next pita.
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When the timer beeps, flip the pita in the oven with a long-handled spatula and set the timer for 1 minute more. Or, throw in the next pita and set the timer for 2 minutes, and then watch the first one around 1 minute in to see if it is ready. Each pita will need between 3 and 4 minutes total baking time. They should be very slightly browned and puffed up when they're done.
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Remove the pitas to a bread basket or plate with several dry clean dish cloths to keep them wrapped and warm.
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Place everything on the table and serve when the pitas are finished baking.