For the dough: Combine the warm water, olive oil, yeast, sugar, and salt in a 5-quart bowl; preferably, in a lidded (not airtight) plastic container or food-grade bucket. Mix until all of the flour is incorporated using a stand mixer or dough whisk. Cover, and allow to rise at room temperature for 2 hours. You can use the dough right away or refrigerate it for up to 14 days.
To Grill the Pizza: Heat your gas grill: Place a baking stone on the primary burners. Turn all burners to high and let heat up for 20 minutes. After they have heated, turn the side without the stone down to low heat.
While your grill is heating, pull out a 10-ounce piece of dough from your bucket and quickly form it into a ball. Let it sit on the counter while you gather your toppings.
Roll the ball out into a 1/8-inch-thick round, approximately 12-inches in diameter. If the ball is resisting, let it rest for about 5 minutes and it will relax and allow you to work with it more easily.
Using a floured pizza peel, place the shaped pizza dough over the pizza stone. Let it cook there until the top starts to bubble and the bottom creates a char to your liking. Remove the pizza from the grill and place on a nearby work surface. Making sure the charred-side is up, top your pizza: cover the pizza with sauce, veggies, and then the cheese.
Then, using your pizza peel, bring the pizza back to the grill, and finish cooking. Place over the hot side again, keeping a very careful watch. As soon as your char-marks look great, slide the pizza over to the cool side and cover the grill. Let cook for 4 to 10 minutes, until the cheese has melted. Remove the pizza from the grill, move to a wire rack, and let cool for a minute or two. Slice into pieces and serve.