1. In the bowl of an electric stand mixer, combine the water, yeast, and sugar. Stir briefly and let stand for 5 minutes or until the top surface has a thin, frothy layer (this indicates that the yeast is active). Add the flour, oil, and salt. Fit the mixer with the dough hook and mix on low speed for about 1 minute or until the dough begins to come together. Increase the speed to medium. Continue to mix until the dough is slightly sticky, smooth, and elastic, about 10 minutes. Form the dough into a ball and place in a lightly oiled bowl. Turn it over to coat all sides and tightly cover the bowl with plastic wrap. Allow the dough to rise in a warm place until it has doubled in size, 1½ to 2 hours.
2. In a large bowl mix ¼ cup oil with the salt and pepper. Place the garlic cloves in the middle of a small sheet of aluminum foil, about 8 inches square, and pour 1 tablespoon of the oil mixture over the garlic. Fold up the sides to make a sealed packet, leaving a little room for the expansion of steam.
3. Add all the peppers and mushrooms to the bowl with the remaining oil mixture. Toss to coat the vegetables evenly.
4. Prepare the grill for direct cooking over medium-high heat. Preheat a perforated grill pan over direct medium-high heat for about 10 minutes. While the pan is preheating, place the packet of garlic over direct medium-high heat and cook until the cloves are soft and light brown, 15 to 20 minutes. Remove the packet from the grill. When the grill pan is really hot, using tongs, lift the mushrooms and peppers from the bowl and spread them out in a single layer on the pan. Cook until they are nicely browned and tender, about 6 minutes, stirring once or twice. Wearing insulated barbecue mitts, remove the pan from the grill and set it down on a heat-proof surface.
5. Cut the dough into 8 equal pieces. Lightly brush eight, 9-inch squares of parchment paper on one side with oil. Using your fingers, flatten each piece of dough on a sheet of parchment paper to create 8 rounds. Each round should be about 1⁄3 inch thick and 6 to 8 inches in diameter. Lightly brush the tops with oil. Let the rounds sit at room temperature for 5 to 10 minutes.
6. Lower the temperature of the grill to medium heat. Brush the cooking grates clean. Working with 4 rounds at a time, place the dough on the cooking grate with the paper sides facing up. Grill over direct medium heat, with the lid closed as much as possible, until the dough is well marked and firm on the underside, 2 to 5 minutes, rotating as needed for even cooking. Peel off and discard the parchment paper. Transfer the crusts to a work surface with the grilled sides facing up. Repeat with the other 4 rounds.
7. Spread the cheese evenly over the crusts and then arrange some garlic, mushrooms, and pepper slices on top. Return the pizzas to the grill and cook over direct medium heat, with the lid closed as much as possible, until the cheese is melted and the bottom of the crusts are crisp, 2 to 5 minutes, rotating the pizzas occasionally for even cooking. Transfer to a cutting board and cut into wedges. Serve warm.