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