Begin by making the dough so it has time to rest and rise. In the bowl of a standing electric mixer fitted with a dough hook, or in a mixing bowl, combine the yeast, sugar and warm water and stir gently to dissolve. Let the mixture stand until the yeast comes alive and starts to foam, 5 to 10 minutes.
If you're using a mixer, turn the mixer on low, add the salt and 2 tablespoons of olive oil. Add the flour, a little at a time, mixing at the lowest speed until the flour has been incorporated. When the dough starts to come together, increase the speed to medium and let it go until the dough gathers into a ball. This should take about 5 minutes. Stop the machine periodically to scrape the dough off the hook. Get a feel for the dough as you're making it by squeezing a small amount together between your thumb and fingers. If it's crumbly, add more water, if it's sticky, add more flour, 1 tablespoon at a time. 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 put it into a lightly oiled bowl, turning it over to coat the dough entirely with the oil. 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.
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 the 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.
Now prepare the topping. Cut up the tomatoes and add to a large mixing bowl. Add the garlic and basil. Crumble in the sausage. Rip the mozzarella into pieces with your hands and add that too along with some extra-virgin olive oil. Season with salt and pepper, add the salad mix and toss to combine.
Once the crust is set and cooked, transfer it to a platter. Scoop up the salad mix in your hands and spread it evenly over the top. Sprinkle with a good dusting of Parmigiano-Reggiano and finish with a drizzle of olive oil.