Grilled pizza was popularized by George Germon and Johanne Killeen at their restaurant Al Forno in Providence, Rhode Island. The couple had the brainstorm to toss a sheet of raw pizza dough on the grate of their charcoal-burning grill.
Step 1: Combine the yeast, sugar, and water in a small bowl and stir to mix. After 5 to 10 minutes, the mixture should look foamy. Place the flour and salt in the bowl of a food processor fitted with a plastic dough blade or a metal blade and pulse to mix. Add the yeast-water mixture and 2 tablespoons of oil and run the machine in short bursts to obtain a smooth, soft dough. It should be moist but not sticky. If necessary add more flour a spoonful at a time, running the machine between additions, until the dough is the proper texture. You can also make the dough in a mixer fitted with a dough hook: Place the yeast, sugar, and water in the mixer bowl. When foamy, add 2 tablespoons oil, and the salt and flour. Mix at low speed to form a smooth, soft dough.
Step 2: Turn the dough out onto a work surface and knead it by hand for a few minutes. Lightly coat a large bowl with the remaining oil. Place the dough in the bowl, turning it to coat with oil. Cover the bowl with plastic wrap and let the dough rise in a warm spot until doubled in bulk, 1 to 1-1⁄2 hours.
Step 3: Punch down the dough and let rise until doubled in bulk again, 1 hour. This second rising isn’t absolutely imperative, but your crust will be lighter if you have the time to do it.
Step 4: Meanwhile, prepare the topping: Rinse the arugula and pat dry with paper towels. Heat 2 tablespoons of oil in a non-stick frying pan over medium heat. Fry the garlic until golden brown, 2 minutes. Transfer the garlic with a mesh skimmer to paper towels to drain. Add the onion to the pan. Cook over medium heat until caramelized to a dark golden brown, 6 to 10 minutes, adding salt and pepper to taste. You may need to lower the heat as the onion darkens. Transfer the onion to a platter. Heat the pan over high heat, adding the remaining 1 tablespoon oil if needed. Blot the tomato slices dry on paper towels and season with salt and pepper. Quickly fry the tomato slices on both sides. Add the garlic and tomatoes to the platter with the onion.
Step 5: Set up the grill for three-zone direct grilling and preheat the hot zone to high. When ready to cook, brush and oil the grill grate.
Step 6: Generously oil two rimmed baking sheets. Place half the dough on one of the baking sheets and stretch it out to form a 9-by-13-inch rectangle that is 1⁄8 to 1⁄4 inch thick. Repeat with the remaining dough and cover it with plastic wrap.
Step 7: Starting from the far narrow side and using both hands, gently lift the first dough rectangle and drape it onto the grate over the hottest part of the fire. Within a minute or so, the underside of the dough will crisp and darken and the top will puff slightly. Using tongs and a spatula, slide the dough to a cooler part of the grill and cook until the bottom of the dough is browned and firm, 3 to 5 minutes. Brush the uncooked top with oil.
Step 8: Using tongs and a spatula, invert the crust over a hotter part of the fire. Grill until this side, too, starts to crisp and darken, about 1 minute. Move the crust back to a cooler part of the grill and brush the top with oil. Arrange half of the cheese slices on the pizza, followed by half the fried tomato slices, caramelized onion, cooked garlic, olives, and capers. Cover the grill to melt the cheese. When the bottom crust is cooked, browned, and firm, 2 to 4 minutes more, slide the pizza onto a platter for serving. Top with half the arugula and season with salt and pepper. Cut into pieces and serve. While people are enjoying the first pizza, prepare the second one the same way (the pizza should be eaten as soon as it comes off the grill).
Variations: There’s no limit to the toppings for grilled pizza. You could certainly go the mozzarella, pepperoni, tomato sauce route (put the mozzarella on first, then the pepperoni, and finally the tomato sauce, so they cook in the proper order). Another of my favorites is crumbled Roquefort cheese, thinly sliced prosciutto, and grilled figs.