For the dough: In the bowl of a standing electric mixer fitted with a dough hook, combine the yeast, sugar, and warm water; stir gently to dissolve. Let the mixture stand until the yeast comes alive and starts to foam, about 5 to 10 minutes. Turn the mixer on low and add the salt and 2 tablespoons of olive oil. Add the flour, a little at a time, mixing at the lowest speed until all the flour has been incorporated. When the dough starts to come together, increase the speed to medium; stop the machine periodically to scrape the dough off the hook. Get a feel for the dough by squeezing a small amount together: if it's crumbly, add more water; if it's sticky, add more flour - 1 tablespoon at a time. Mix until the dough gathers into a ball, this should take about 5 minutes.
Turn the dough out onto a lightly floured surface and fold it over itself a few times; kneading until it's smooth and elastic. Form the dough into a round and place in a lightly oiled bowl, turning it over to coat. Cover with plastic wrap or a damp towel and let it rise in a warm spot (i.e. over a gas pilot light) until doubled in size, about 1 hour.
Once the dough is domed and spongy, turn it out onto a lightly floured counter. Roll and stretch the dough into a cylinder and divide into 3 equal pieces. Cover and let rest for 15 minutes so it will be easier to roll out. In the meantime, make the pizza topping.
For the topping: Coat a saute pan with the olive oil and place over medium heat. When the oil gets hazy, add the onion and garlic, cook and stir for 5 minutes to soften. Stir in 1/2 bunch torn basil leaves. Carefully add the tomatoes (nothing splashes like tomatoes), cook and stir until the liquid is cooked down and the sauce is thick, about 10 minutes; season with salt and pepper and then take it off the stove.
Preheat an outdoor gas or charcoal barbecue and get it very hot. Brush the grates with oil to keep the sausages from sticking. Grill the sausages until cooked through, turning with tongs. Set aside. Keep the grill hot to make the pizza.
Roll or pat out a piece of dough into a rustic, oblong shape, about 1/4-inch thick – do this directly on a lightly floured pizza paddle for ease. Brush the crust with a thin layer of olive oil, and lay oil-side down onto the hot grill. Once the dough looks set, in about 2 minutes, turn the crust over. Spread the tomato sauce evenly on the crust with a ladle. Rip the mozzarella into pieces with your hands and distribute them on top; crumble the grilled sausage all over the pizza. Close the grill cover and cook until the cheese is melted and bubbling and the crust is golden and crisp, about 5 minutes. Remove the pizza from grill and sprinkle with some fresh basil leaves and a good dusting of Parmesan. Repeat with the remaining dough.