Special equipment: 8 pieces of parchment cut to fit a baking sheet
Make the pizza dough: Whisk together the bread flour and salt in a large bowl. Combine the water, oil and sugar in a second large bowl; add the yeast, and let sit until it begins to foam, about 5 minutes. Add the flour mixture to the yeast mixture, and use your hands to combine until it comes together into a rough, shaggy ball. Cover the bowl with a kitchen towel and set aside for 15 minutes.
Sprinkle a work surface with flour and knead the dough until it is smooth and elastic, about 5 minutes. Clean the bowl and lightly coat it with oil. Return the kneaded dough to the bowl, brush with oil, cover and set aside in a warm place until doubled in size, about 1 hour.
Transfer the dough to the work surface, and punch it down. Quarter the dough, and cut each quarter into 4 for a total of 16 pieces. Shape each into a ball. Evenly space the dough balls on the work surface, cover and let sit until they poof slightly, about 30 minutes. (To make ahead, after cutting the dough, wrap each piece tightly in plastic wrap and refrigerate overnight, then let sit at room temperature for 30 minutes before rolling.)
Prepare a grill for medium-high heat and brush the grill grates with oil.
Roll out the dough: Line a baking sheet with a piece of parchment, and sprinkle with cornmeal to keep the dough from sticking. Roll out each ball of dough into an 8-inch circle on the work surface (keep the other balls covered as you work). Arrange 2 to 3 rounds without overlapping on the parchment. Repeat with layers of parchment, cornmeal and dough rounds.
Par-grill the dough: Working in batches, brush the top of each dough round with oil and sprinkle with salt. Grill, oiled-side down, until grill marks just form and the dough starts to bubble up, 30 seconds to 1 minute. Turn the dough over and repeat. Transfer to a cooling rack or another baking sheet to cool. The dough shells will be underbaked and flimsy (they finish cooking when guests top and grill them). Stack the shells, wrap tightly in plastic wrap and set aside until ready to use. (The shells can be refrigerated for up to 2 days. Or tightly wrap each one in plastic and foil, then freeze for up to 3 months; thaw at room temperature for a few minutes before topping and grilling.)
Set up the pizza bar: Prepare a grill for medium-high heat and brush the grill grates with oil. Set out a large cutting board with a knife or pizza cutter nearby. Set out a stack of the pizza shells and any combination of the suggested sauces, cheeses and toppings.
Grill the pizzas: Let your guests add their desired toppings to the dough shells. Transfer the pizzas to the grill, cover and cook until the bottom of each is slightly charred, the cheese has melted and the toppings are cooked through. Remove the pizzas from the grill, slice them and encourage your guests to top them with some of the finishings.
One 15-ounce can tomato sauce
For the red sauce: Combine the tomato sauce, basil, oil, garlic and 3/4 teaspoon salt in a small bowl. Cover and refrigerate for at least 1 hour before using. The sauce can be made up to 1 day ahead.
For the arugula-pistachio pesto: Pulse the arugula, basil, parsley, pistachios, capers, 3 tablespoons water and 1/2 teaspoon salt in a food processor until a paste. The sauce can be made up to 1 day ahead; refrigerate.
For the ricotta spread: Stir together the ricotta, Parmesan, oil, garlic and 1/2 teaspoon salt in a medium bowl. The sauce can be made up to 1 day ahead; refrigerate.
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