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Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess. Mix the biga ingredients in a small bowl until well combined. Let the biga rest overnight, covered, or for up to 15 hours.
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Place all of the dough ingredients, including the biga, into the bowl of your mixer, and beat it at medium speed (speed 4 on a KitchenAid), using the flat beater, for 3 minutes; the dough should have started to clear the sides of the bowl. If it hasn't, increase the speed of the mixer, and mix for an additional 2 minutes.
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Switch to the dough hook, and knead for an additional 3 to 4 minutes; the dough will be extremely soft.
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Transfer the dough to a lightly greased bowl, cover it, and allow it to rise for 2 hours, deflating it once midway through the rising time.
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Transfer the dough to a well-oiled work surface. Lightly grease a half-sheet baking pan (18" x 13") or similar large pan, and your hands.
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Using a bench knife or your fingers, divide the dough in half. Handling the dough gently, stretch one piece into a log about 10" long, and place it crosswise on one half of the baking sheet. Flatten the log with your fingers until it's about 3" wide.
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Repeat with the remaining piece of dough. Lightly cover the dough with heavily oiled plastic wrap, and allow it to rise for 2 hours; it'll become very puffy, and will jiggle when you shake the pan gently. Towards the end of the rising time, preheat the oven to 425°F.
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Bake the loaves for 22 to 25 minutes, or until they're golden brown.
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Turn off the oven, remove the panini loaves from the baking sheet, and return them to the oven, propping the oven door open a couple of inches with a folded-over potholder. Allow the loaves to cool completely in the oven; this will give them a very crisp crust. If desired, sift a dusting of flour on top of the cooled loaves.