For the dough, combine all ingredients in a food processor or by hand. Cover with plastic wrap and let rise to desired size.
Shape the pizza dough by flattening it with your hands on a slightly floured surface. Either use your fingers to stretch the dough out, or hold up the edges of the dough with your fingers, letting the dough hang and stretch, while working around the edges of the dough. Once you've stretched the dough, let it sit for 5 minutes and then push out the edges with your fingers again, until you have a nice round shape, about 12-inches in diameter. Do not make a raised rim, it will interfere with the grilling process.
Once the grill is hot, place a pizza dough round on a lightly floured (or you can use cornmeal) rimless cookie sheet (or pizza peel if you have one). Let the dough slide off the cookie sheet onto the hot grill grates. Close the lid of the grill and let cook for 2 minutes.
After 2 minutes, open the grill and check underneath the dough to see if it is getting browned. Once the pizza dough has browned lightly on one side, use your cookie sheet or pizza peel to remove it from the grill. Use a spatula to flip the dough over so that the grilled side is now up.
Brush with tomato sauce and top it with the remaining ingredients.
Slide the topped pizza back onto the grill. If you are using a gas grill, reduce the heat. If working with a charcoal grill, close the vents on the cover almost all the way. Close the lid and cook for 2-3 minutes more, or until the bottom begins to char and the cheese is bubbly.