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When ready to cook the pizza, combine in a blender or NutriBullet all of the ingredients for the whipped ricotta. Add the remaining garlic oil and blend until the ricotta mixture is whipped and smooth, about a minute. Stir once and blend again for 20 seconds until everything is well blended.
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Reduce the grill temperature to 400 degrees so the bottom of the pizza does not burn. Open the grill to bring the temperature down.
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To assemble the pizza, spread the whipped ricotta mixture all over the cooked pizza dough. Top with the figs. Sprinkle the fontina and taleggio over the pizza and top with the prosciutto. Break the burrata into chunks with your fingers and dollop evenly over the pizza.
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Slide the pizza back onto the heated pizza stones. Close the grill and cook the pizza for about 5 minutes, making sure that the bottom crust does not burn. Remove the pizza from the grill and serve topped with the arugula.