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Prepare the grill and let burn down to medium coals or heat gas grill on medium heat.
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or if using oven... preheat oven to 400F (200C).
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In a colander set over another bowl, salt the eggplant and let drain for 30 minutes then rinse and dry.
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Brush the eggplant slices with olive oil and season with cayenne pepper.
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Toss the peppers with a little olive oil.
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Place both on the grill and cook the eggplant until browned on the first side, about 3 minutes.
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Note: If using stove use burner of gas stove.
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If using oven to blacken; use broiler of oven by placing peppers on cookie sheet and placing under broiler until blackened.
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The slices should be browned between the grill marks from outside grill when ready to turn.
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Move them away from direct heat and cook on the second side until tender, about 10 minutes total.
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Cut the grilled slices into 1/2-inch dice.
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Cook the peppers, turning as necessary, until charred all over.
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Remove the peppers to a bowl.
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Cover and let the peppers steam to loosen the skins.
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When cool enough to handle, peel, seed, and tear into long, thin strips.
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Divide the dough into 6 equal balls.
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Working on a surface free of flour, roll each ball under your palm.
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As it rolls, it will stick slightly to the surface, creating tension that helps form a tight, round ball.
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Dust the work surface lightly with flour, pat each ball down lightly, dust the tops with flour, cover with a towel, and let rise for about 15 minutes.
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With a rolling pin, roll out each ball into a circle 8 or 9 inches in diameter and about 18 inch thick.
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Have the grill ready at medium heat.
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Have the 1/4 cup olive oil, eggplant, peppers, garlic, oregano, red pepper flakes, if using, olives, and cheese in bowls and ready by the side of the grill.
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Brush each dough round with about 1 teaspoon of the olive oil.
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Place on the grill, oiled side up, and cook until browned on the underside, about 2 minutes.
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Turn and, working quickly, brush each round with another 1 teaspoon olive oil.
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If using inside oven, place dough on a cookie sheet sprinkled with a little yellow corn meal.
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Then scatter each with 1/6 each of the garlic, oregano, red pepper flakes, if using, eggplant, bell peppers, olives, and cheese.
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Cover the grill and cook until the rounds are browned on the bottom and the cheese has softened, about 7 minutes.
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Transfer to plates and top each with an equal amount of the arugula.
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Spritz lightly with vinegar.
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Serve immediately.
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Directions For Pizza Dough:
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In a small bowl, combine the yeast, sugar and water and stir gently to dissolve.
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Let the mixture stand until the yeast comes alive and starts to foam, 5 to 10 minutes.
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If you're using a stand mixer, combine the salt and flour to the bowl and pulse a few times to mix.
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Add the yeast mixture, at the lowest speed, until the flour incorporates.
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When the dough starts to come together, increase the speed to medium and mix until the dough gathers into a ball.
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This should take about 2 minutes.
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Add the olive oil and pulse a few more times.
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Stop the machine periodically to scrape the dough off the hook.
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Get a feel for the dough as you're making it by squeezing a small amount together between your thumb and fingers.
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If it's crumbly, add more water, if it's sticky, add more flour, 1 tablespoon at a time.
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Turn the dough out onto a lightly floured surface and fold it over itself a few times, kneading until it's smooth and elastic.
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*If you're making the dough by hand, add the yeast mixture to a large bowl and stir in the salt and the 2 tablespoons of olive oil.
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Then begin stirring in the flour.
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When the mixture becomes too stiff to stir with a spoon, knead in the rest of the flour by hand, adding just enough so that the dough is soft but not too sticky.
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As you work, squeeze a small amount of dough together between your thumb and fingers.
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If it's crumbly, add more water; if it's sticky, add more flour, 1 tablespoon at a time.
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Knead until smooth and elastic, about 10 minutes.
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Form the dough into a round and put it into a lightly oiled bowl, turning it over to coat the dough entirely with the oil.
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Cover with plastic wrap or a damp towel and let it rise in a warm spot (over a gas pilot light) or in a warm area in kitchen until it doubles in size, about 1 hour.