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FOR THE CRUST:.
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Combine oil and water in liquid measuring cup. I food processor fitted with plastic dough blade or metal blade, process bread flour, whole wheat flour, sugar, salt and yeast until combined, about 5 seconds. With machine running, slowly add liquid through feeding tube; continue to process until dough forms tacky, elastic ball that clears sides of work bowl, about 1 1/2 minutes. If dough ball does not form, add more flour 1 tablespoons at a time and process until dough ball forms. Spray medium bowl lightly with nonstick cooking spray or rub lightly with oil. Transfer dough to bowl and press down to flatten surface; cover tightly with plastic wrap and set in draft-free spot until doubled in volume, 1 1/2 - 2 hours.
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When dough has doubled, press down gently to deflate; turn dough out onto work surface and divide into 4 equal-sized pieces. With cupped palm, form each piece into a smooth, tight ball. Set dough balls on well-floured work surface. Press dough rounds with hand to flatten; cover loosely with plastic wrap and let rest about 15 minutes.
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FOR THE TOPPING:.
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Meanwhile, toss tomatoes and table salt in medium bowl; transfer to colander and drain 30 minutes (wipe out and reserve bowl). Shake colander to drain off excess liquid; transfer tomatoes to now-empty bowl and set aside. Combine cheese in second medium bowl and set aside.
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Gently stretch dough rounds into disks about 1/2 inch thick and 5-6 inches in diameter. Working one piece at a time and keeping the rest covered, roll out each disk to 1/8 inch thickness, 9-10 inches in diameter, on well-floured sheet of parchment paper, dusting with additional flour as needed to prevent sticking, (If dough shrinks when rolled out, cover with plastic wrap and let rest until relaxed, 10-15 minutes). Dust surface of rolled dough with flour and set aside. Repeat with remaining dough, stacking sheets of rolled dough on top of each other (with parchment in between) and covering stack with plastic wrap; set aside until grill is ready.
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SPICY GARLIC OIL:.
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Combine the olive oil, minced garlic, onion, salt and chili in a small saucepan - saute for two minutes over medium - low heat. Double or treble this recipe to have lots made ahead of time!
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TO GRILL:.
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Light all burners and then to high heat, cover grill, and heat grill until hot, about 15 minutes; scrape cooking grate clean with grill brush.
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Lightly flour pizza peel; invert 1 dough round onto peel gently stretching it as needed to retaing shape (do not stretch dough too thin; thin spots will burn quickly). Peel off and discard parchment; carefully slide round onto hot side of grill. Immediately repeat with another dough round. Cook (with grill lid down) until tops are covered with bubbles (pierce larger bubbles with paring knife) and bottoms are grilled marked and charred in spots, 2-3 minutes; while rounds cook, check undersides and slide to cool area of grill if browning too quickly. Transfer crists to cutting board browned sides up. Repeat with 2 remaining rounds.
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Brush 2 crusts generously with Spicy Garlic Oil; top each evenly with on-quarter of cheese mixture and one-quarter of tomatoes. Return pizzas to grill and cover grill with lid; cook until bottoms are well browned and cheese is melted, 3-5 minutes, checking bottoms frequently to prevent burning. Transfer pizzas to cutting board; repeat with remaining 2 crusts. Sprinkle pizzas with basil and coarse salt to taste; cut into wedges immediately.
