Make the dough
Mix 3/4 cup of warm water (between 100°F and 110°F) with the yeast and sugar in a 1-cup measure, and let sit for 5 to 10 minutes, until the surface of the liquid becomes frothy. In a stand mixer fitted with a dough hook, mix the flour,1 Tbs. oil, and 1 tsp. salt on low speed until combined. Add the yeast mixture and continue mixing, scraping down the sides of the bowl as necessary, until a dough forms, about 3 minutes. Add 1 or 2 Tbs. of water or flour if the dough is too dry or wet, respectively. Transfer the dough to a floured work surface and knead until it becomes smooth and elastic, about 5 minutes. Transfer to a large oiled bowl, cover with a dishtowel, and let sit at room temperature until the dough almost doubles in size, 1 to 2 hours. (The dough can be made up to 2 days ahead and refrigerated in a zip-top plastic bag.)
Make the pizzas
For a gas grill heat the back burner(s) to medium-high and the front burner to low. For a charcoal grill, light a large fire and push the coals to one side of the grill; the fire is ready when you can hold your hand a couple inches above the hot zone for just 3 to 4 seconds. Clean and oil the grill grates. Brush both sides of the eggplant slices with 2 Tbs. oil, sprinkle with 1/2 tsp. salt, and grill over the hot part of the fire along with the sausage, flipping occasionally, until the eggplant is completely tender, about 8 minutes, and the sausage is cooked through, 10 to 12 minutes. Let cool for a couple minutes and then thinly slice the sausage. In a medium bowl, mix the tomatoes with the thyme, 1/4 tsp. salt, and 1/2 tsp. black pepper.
Line two baking sheets with parchment paper. Form the dough into 2 equal balls and transfer to a floured work surface. Shape each into a thin round, 10 to 12 inches wide. Brush the top side with 1 Tbs. oil then flip the oiled side over onto the prepared baking sheet. Brush the other side with 1 Tbs. oil and cover with a piece of parchment paper. Repeat with the second ball of dough and the second baking sheet. (You can grill the pizzas immediately or hold at room temperature for up to 1 hour.)
When you are ready to grill, peel back the parchment on top of one dough round. With two hands, support the dough under the remaining parchment paper side, flip it dough side down onto the grill’s hot zone, and peel off the parchment paper from the facing side. Cook the dough without moving until it bubbles and gets good grill marks, about 1 minute. Rotate it 90° and cook until the dough is uniformly browned but not burnt, about 30 seconds to 1 minute. Pull the dough to the cooler zone of the charcoal fire or to the front of the gas grill and reduce the middle zone’s heat to medium-low. Flip the dough and top evenly with half of the tomatoes, sausage, eggplant, mozzarella, Parmigiano-Reggiano, a drizzle of the remaining Tbs. of oil, and a light sprinkling of salt. Cover the grill (with the vents open on a charcoal grill) and cook, rotating the pizza about every minute until it cooks evenly and the cheese melts, about 3 to 5 minutes. Sprinkle with the basil, transfer to a large cutting board, slice, and tent with aluminum foil while cooking the second pizza. Grill and top the remaining pizza in the same manner, and serve.