To make the crust: Sprinkle yeast over 3/4 cups warm water. Let stand for a few minutes.
In a mixer, combine flour,garlic,cheese and salt. With the mixer running on low speed (with paddle attachment), drizzle in olive oil until combined with flour. Next, pour in yeast/water mixture and mix until just combined, and the dough comes together in a sticky mass.
Coat a separate mixing bowl with a light drizzle of olive oil, and form the dough into a ball. Toss to coat dough in olive oil, then cover the bowl tightly with plastic wrap and set it aside for 1 to 2 hours, or store in the fridge until you need it.
To make the pesto: Add basil leaves, 1/2 cup Parmesan, pine nuts, and salt and pepper to a food processor or blender. Turn machine on, then drizzle in olive oil while it mixes. Continue blending until combined, adding additional olive oil if needed. Set aside.
To make the toppings: Brush zucchini and mushrooms slices with olive oil. Sprinkle with salt and pepper. Grill over medium heat until vegetables are tender and have nice grill marks, sprinkle with minced garlic and set aside.
Place whole peppers on grill and allow to blacken. Remove from grill and immediately place in a large ziploc bag. Allow to sit for 20 minutes or so, then remove peppers from bag and peel off blackened skin. Deseed peppers, then slice into strips. Season with salt and pepper. Set aside.
To assemble the pizza:Heat a large grill pan over medium high heat. Drizzle with olive oil.
Divide dough into 4 pieces,roll out each piece on a floured surface; get it as thin as you can.
Place pizza crust on grill pan and cook until done,( the dough will puff up a bit and look dry) turn over cook on other side.
Repeat until all pizza crusts are cooked.
Spread a generous layer of pesto all over the surface of the crusts. (You might have a little left over; if you do, store in the fridge.) Divide the toppings into four piles.
Lay slices of mozzarella all over the pesto. Arrange grilled zucchini, mushrooms, and peppers in a pretty pattern on top of the mozzarella. Place in a hot broiler just until cheese melts.
Remove from oven and crumble with Parmesan cheese and sprinkle with basil leaves.