Warm 1 cup of the water and proof the yeast in it until it bubbles.
In a small bowl, mix the old dough with 1 cup of cool water and 4 or 5 handfuls of flour, stirring until you have a pastry mixture. Dissolve the honey or malt extract in the yeast mixture and add this to the old dough mixture.
Place half of the rest of the flour and half of the salt in a food processor and pulse to mix. Add half of the old dough, yeast, and malt extract mixture. Pulse the processor to combine the ingredients, then slowly add half of the remaining water (1/4 cup) through the feed tube while the processor is running. Process the mixture for between 30 seconds and one minute. The dough should be wet and sticky. In the last 10 seconds of processing, pour half of the olive oil through the feed tube.
Empty the dough into a large bowl and combine the remainder of the ingredients in the food processor to make a dough in the same way.
Combine both batches in the bowl by mxing them together with a wooden spoon or plastic dough scraper.
Let the dough rise, covered, in a warm place for 1 hour.
The dough will still be very wet, so sprinkle plenty of flour over it and remove it from the bowl, turning it out onto a well-floured worktable.
Divide into 10 pieces about the size of tangerines (each will weigh about 4 to 5 ounces) for individual pizzas or calzone. Round the dough up into tight little balls. Cover the balls with plastic wrap and let them rise a second time for between 3 and 5 hours. When ready the dough balls will be very puffy but, because they are so moist, will be somewhat flat.
Flatten each piece of dough with the palm of your hand. With your fingertips, poke the dough in the middle, leaving a 1/2-inch strip around the edge untouched, and making sure you do not poke through the dough. Lift the piece up, place it on the curled-up knuckles of both hands and gently stretch, while rotating the piece. The result will be a flat disk that measures 1/8 inch thick and 6 or 7 inches in diameter.
Cover with your favorite sauce and toppings and bake on a preheated baking stone at between 450 degrees and 500 degrees for 5 to 8 minutes or until the bottom crust is crisp and the top is bubbling.
*Dough that has been kept at a room temperature should be between 6 and 8 hours old. If kept in the refrigerator, it should be 24 hours old.
**If the old dough contains salt, use only 1 tablespoon salt in this recipe.
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