How to Make It
In a food processor, pulse the pistachios and garlic until minced. Add the arugula, basil and crushed red pepper and pulse until minced. With the machine on, gradually add the 1/2 cup of oil until incorporated. Add the Parmigiano and pulse to combine. Season the pesto with salt and black pepper.
Light a grill or preheat a grill pan. In a bowl, toss the mozzarella with the Fontina, provolone and scamorza.
Divide the dough into 4 pieces. On a lightly floured work surface, roll or stretch each dough piece to a 10-inch oval, 1/8 inch thick. Brush the dough with oil. Grill each pizza over moderately high heat, turning once, until lightly charred on both sides and puffed, about 4 minutes. Transfer to a work surface. Spread one-fourth of the pesto on top and sprinkle with the cheeses. Season with salt and pepper.
Working in batches if necessary, grill the pizzas over moderately low heat, covered, until the cheese is just melted, about 5 minutes. Cut the pizzas into strips and serve hot.