Prepare the dough: Whisk 1 cup warm water (about 105 degrees F) and the sugar in a small bowl and sprinkle the yeast over top. Let stand until foamy, about 10 minutes. Stir in the olive oil.
Whisk the flour and salt in a large bowl. Make a well in the center, pour in the yeast mixture and stir to make a shaggy dough. Turn out onto a floured surface and knead until smooth and elastic, about 5 minutes, adding just enough flour to prevent it from sticking. Form into a ball, put in a large oiled bowl and turn to coat with the oil. Turn seam-side down, cover with plastic wrap and let rise at room temperature until doubled, about 1 hour 30 minutes. Meanwhile, preheat a grill to medium high (if using a charcoal grill, bank the coals to one side) and grill the toppings (see below).
Arrange the grilled vegetables in piles on a baking sheet, season with salt and pepper, and drizzle with olive oil. Add the remaining toppings to the baking sheet; place near the grill.
Reduce the grill heat to medium low (on a charcoal grill, spread the coals into a single layer and let cool until they are covered in a light gray ash). Divide the dough in half and dust the dough and work surface with flour. Roll each piece into a 7-by-12-inch rectangle, about 1/8 inch thick. Brush an inverted baking sheet with olive oil and lay both crusts on it. Lift each piece by the corners and lay on the grill, oiled-side down; cook until the dough puffs and the underside is stiff and marked, about 5 minutes.
Flip the dough with tongs, brush with olive oil, spread with the tomato sauce and top with mozzarella. Arrange the grilled vegetables and roasted peppers on top. Cover and cook until the cheese melts, about 5 more minutes.
Transfer the pizzas to a cutting board. Top with the capicola, greens, basil, a drizzle of olive oil and salt.
Grilled Pizza Toppings:
Tomato Sauce Grill 3 vine-ripened tomatoes, turning, until charred, 10 minutes. Transfer to a bowl, add 1 chopped garlic clove, 2 tablespoons olive oil, and salt and pepper. Cool, then crush into a sauce.
Onion/fennel Cut 1 medium red onion or 1 small fennel bulb into 8 wedges. Toss with 1 tablespoon olive oil and season with salt and pepper. Grill, turning, until soft, about 10 minutes; slice.
Mushrooms Toss 8 ounces stemmed shiitake mushrooms with 3 tablespoons olive oil and season with salt and pepper. Grill, turning, until soft, about 2 minutes per side; cut in half.