In a large bowl, whisk together water and yeast; let stand for 5 minutes. Whisk in salt, garlic powder and olive oil. Stir in about 3 1/2 cups of the flour, or until a soft dough forms. Sprinkle the work surface with the remaining flour, turn out the dough, and knead until smooth, about 5 minutes. Lightly oil a large bowl. Put in the dough, turn it too coat it with oil, and cover with a clean towel. Let the dough rise until it is puffy and light — about 2 hours.
The dough can be prepared to this point up to 3 days in advance. Return it to room temperature before using. The dough can be frozen at this point for up to 2 months.
After kneading, punch dough down, divide it into fourths if desired, and wrap it tightly. Thaw it in the refrigerator overnight, then set it out, coated with oil, to rise at room temperature.
Preparing the Pizzas
Light a charcoal fire and let it burn until the coals are evenly white or preheat a gas grill. Wrap wood chips partially in foil, creating a small packet that is open at the top. Set the packet on the hot coals or hot lava stones. Position the grill rack about 6 inches above the heat source.
Pat one piece of pizza dough to a 12-inch diameter round on a floured working surface. Transfer the dough to a grill, placing it on the cool zone. Cover the grill and cook the dough until it is puffed on top and lightly marked by the grill, but not at all crisp on the bottom, 4 to 5 minutes.
Transfer the crust to a work surface and flip it over. Spread 1/3 cup tomato sauce over the dough, leaving a 1/2-inch border. Scatter cheese over the tomato sauce. At this point, add roasted, peeled peppers, grilled vegetables, or any other topping you desire. Return the pizza to the grill, cover, and bake, rotating it once or twice to ensure even browning. It is done when the cheese is melted and bubbling, the vegetables are heated, and the crust is cooked through and lightly crisped or even charred a little bit (about 10 minutes).
Transfer the pizza to a cutting board and let stand for 5 minutes before cutting. Repeat with remaining ingredients. Rake in additional coals if the grill becomes too cool for the third and fourth pizzas.