1.Make the dough: Combine the yeast, sugar, and warm water in a measuring cup; let sit for 5 mins. Combine the bloomed yeast, four, and salt in the bowl of a food processor. Pulse to combine. Add the room-temperature water in a steady stream, mixing until the dough begins to form a ball. Tranfer to a floured countertop, and knead with the heel of your hand until the dough is smooth and elastic, about 5 mins.
2. Coat a bowl with the olive oil. Place the dough in the bowl, and cover with a clean damp cloth. Put in a warm spot to rise until double in size, about 1 hour.
3. When the dough has doubled in size, punch the dough, scrape it onto the counter, and knead it lightly into a smooth ball. Cut into 8 pieces and, with a rolling pin, roll each one out to form a very flat 5 to 6 inch circle.
4. Preheat the grill to medium-high. Lighty oil. Place the bread rounds on the hot grill, and cook without touching until you see bubbles on the surface, about 2 mins. Remove from the heat, and add the first 7 toppings. Reduce the heat to medium-low.
5. Return the flatbreads to the grill, close the lid, and cook for 3 minutes, until the cheese is melted and the bottoms are slightly charred.
6.Remove from the heat, and sprinkle with the freshly grated Parmesan and torn basil leaves.
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