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For the calamari: In a large heavy-bottomed saucepan, pour enough oil to fill the pan about a third of the way. Heat over medium heat until a deep-frying thermometer inserted in the oil reaches 350 degrees F. Dredge the calamari in the semolina flour and sift with large mesh strainer to remove any excess flour. Fry the calamari in small batches for 3 to 4 minutes. Season with salt.
For the pizza: Prepare a hot charcoal fire, setting the grill rack 3- to 4-inches above the coals. Grease an inverted baking sheet with some of the oil.
Spread and flatten the pizza dough with your hands into a 10- to 12-inch free-form circle, 1/8-inch thick, on the greased inverted sheet tray. Do not make a lip. You may end up with a rectangle rather than a circle; the shape is not important, but do take care to maintain an even thickness.
When the fire is hot (you can hold your hand over the coals for 3 to 4 seconds at a distance of 5 inches), use your fingertips to lift the dough gently by the two corners closest to you, and drape it onto the grill. Catch the loose edge on the grill first and guide the remaining dough into place over the fire. Within 1 minute the dough will puff slightly, the underside will stiffen and grill marks will appear. Using tongs, immediately flip the crust over, onto the coolest part of the grill. Quickly brush the grilled surface with some of the olive oil. Scatter the fontina and Pecorino Romano over the dough. Spoon 6 tablespoons of the arrabbiata over the cheese but do not cover the entire surface of the pizza with sauce.
Slide the pizza back toward the hot coals, but not directly over them. Using tongs, rotate the pizza frequently so that different sections receive high heat; check the underside often to see that it is not burning. The pizza is done when the top is bubbly and the cheese melts. Top with the calamari, some scallions and an additional drizzle of oil, and serve immediately.