Sift the sugar, yeast, flour and salt in the bowl of a stand mixer. With the mixer running slowly, add the water and olive oil. Continue to mix until the dough forms. Scrape down sides of bowl periodically to ensure everything is well-mixed.
If dough is too dry, add more water.
Lightly grease a bowl with oil. Add the dough and cover with plastic wrap. Let rise for an hour. The dough should double in size.
After an hour, divide the pizza in half. Roll into balls. Cover each with a towel and let rest for 30 to 45 minutes. Lightly flour a work surface. Using a rolling pin, flatten and roll pizza dough into a 12 to 14-inch round shape. Stretch and turn pizza by hand as you go to assist in forming circular shape. Repeat with other dough.
Transfer the dough to a pizza peel (paddle) or wooden cutting board before grilling. Grease the grill of your barbecue with oil. Preheat barbecue to medium.
Brush the dough with olive oil and then gently flip pizza onto the grill. Grill for approximately 2 minutes, turning the pizza a couple times to help it cook evenly.
While the pizza is grilling, brush the other side with another olive oil and immediately flip pizza.
Remove pizza from grill. Add your sauce, toppings and cheese. Return to grill and cook for 1-2 more minutes.