2. Heat some oil in a large sauté pan and cook the peppers for about 5 minutes. Sprinkle with salt and pepper and set aside. Add a bit more oil to the pan and cook the onions over moderate heat for 10 minutes. Set aside. Heat the last of the oil and cook the mushrooms until tender. Sprinkle with salt and pepper and set aside.
(This can be made a day in advance and stored in the refrigerator.)
3. For the dough: Dissolve the yeast and sugar with the warm water. Stir in the 1/2 cup flour and let this mixture rest in a warm place until bubbly. Put the rest of the flour in the bowl of an electric mixer fitted with the dough hook attachment. Make a well in the center and add the bubbly dough mixture, salt, olive oil and water. Mix on low to combine. Continue mixing until the dough leaves the sides of the bowl clean. Turn out into an oiled bowl and let rest, covered with plastic wrap in a warm place until doubled in size.
(The pizza dough can be made a day in advance and stored in the refrigerator. Just allow enough time for it to come to room temperature for easy stretching.)
4. Turn dough onto a lightly floured surface and punch down. Form into 24 balls and let them rest under a cloth for 30 minutes longer.
5. Preheat the oven to 475°F.
6. Roll each piece of dough into a circle about 4-inches in diameter. Brush the pizza rounds with garlic oil and spread with tomato puree. Sprinkle the mozzarella cheese over the tomato puree. Now distribute the seasoned vegetables.
7. Bake 12 to 15 minutes, or until golden. Sprinkle pizza with herbs and Parmesan cheese. Serve at once.