Make the pizza dough:
Combine the bread flour, semolina, yeast, and sea salt in a food processor fitted with the metal blade. Pour 1 cup of cool water through the feed tube, pulsing as you pour. With the motor running, pour the oil through the feed tube and process until the dough comes together, about 1 minute.
Turn the dough out onto a clean work surface—you may or may not need to flour the surface, depending on how tacky the dough is. Knead the dough until it’s smooth and elastic, about 2 minutes, and then gather it into a ball. Grease a large bowl with olive oil and put the dough in the bowl, turning to coat it with oil. Cover the bowl tightly with plastic wrap and leave in a warm place until the dough has doubled in size, 1-1/2 to 2 hours.
Make the pizzas:
Prepare a hot charcoal or gas grill fire. If using charcoal, spread the hot coals across half of the charcoal grate and leave the other half clear. If using gas, turn one of the burners to low to create a cooler zone. Scrub the grill grate with a wire brush and then wipe the grate with a paper towel dipped in oil.
While the grill is heating, shape the pizza dough: Put the dough on a lightly floured work surface. Knead it briefly to expel any air bubbles and then cut it into quarters. Cover 3 pieces with plastic wrap to prevent them from drying out. Using a rolling pin or your hands, flatten the dough into a 10-inch circle about 1/4 inch thick. If at any point the dough resists rolling, set it aside to rest for a few minutes while you work another piece of dough.
Lightly dust a baking sheet with a little semolina or flour. Using your fingers, transfer the rolled-out dough to the baking sheet and cover with parchment. Repeat with the remaining 3 pieces of dough, shingling the dough between lightly dusted sheets of parchment. Cover the baking sheet with plastic wrap and refrigerate until ready to grill the pizzas.
Bring the tomato sauce to a simmer in a small pot over medium heat. Remove it from the heat and stir in the chopped basil. Keep warm.
When the grill is hot, brush the mushroom caps with 2 Tbs. of the olive oil and sprinkle with salt and pepper. Slide toothpicks or skewers horizontally into the onion disks to hold them together. Brush the disks with the remaining 1 Tbs. olive oil and season with salt and pepper. Arrange the mushrooms and onions on the hot side of the grill and cook until grill marks have formed, about 4 minutes. Flip and grill until they are well marked and tender, 3 to 4 minutes more for mushrooms, 1 minute more for onions (if the onions threaten to char, move them to the cool side of the grill). Let cool briefly, remove the skewers from the onions, and then coarsely chop the mushrooms and onions.
Have the pizza dough, sauce, toppings, and a clean baking sheet ready near the grill. Use your fingers to carefully transfer one of the dough rounds onto the grate over the hot part of the grill. Grill just until grill marks form on the bottom and the dough has begun to crisp and puff up a bit, about 2 minutes. (If you are able to fit 2 pizzas on the hot part of the grill at the same time, grill 2 at a time.) Using tongs and a wide spatula, move the crust, cooked side up, to the empty baking sheet. Spoon a thin layer of heated tomato sauce over the surface of the pizza. Scatter 3/4 cup Fontina over the sauce, and then top with one-quarter of the mushrooms and onions.
Using tongs and a wide spatula, slide the pizza, raw side down, onto the cooler side of the grill. Cover and grill until the bottom is browned and crisped and the cheese has melted, 3 to 5 minutes. Using the tongs and spatula, transfer the pizza to a large dinner plate. Top with about 4 tomato slices and lightly season them with salt. Scatter about one-quarter of the arugula and basil over the pizza and drizzle with a little olive oil and balsamic vinegar.
Grill the remaining 3 pizzas in the same way, dividing the toppings evenly among them. Serve as soon as all of the pizzas have been grilled. (If you want the cooked pizzas to remain hot, put them in a 200°F oven as you finish them, but wait until just before serving to top with the tomatoes, arugula, and basil.)