Dissolve yeast in warm water, and add sugar.
Proof for ten minutes, or until frothy. Add salt, olive oil, and flour; mix until dough pulls away from the sides of the bowl.
Turn onto a lightly floured, flat surface.
Knead until smooth, about 6 to 8 minutes. Place dough in a well oiled bowl, and cover with a damp cloth. Set aside to rise until doubled. Punch down, and knead in garlic and basil. Set aside to rise until doubled (1-2 hours).
Preheat grill for high heat. Heat 1/2 cup olive oil with 1 teaspoon minced garlic for 30 seconds in the microwave. Set aside. Punch dough down and divide.
On baking sheets, form into 2 oblong shapes approximately 3/8 to 1/2 inch thick. Brush grate with garlic flavored olive oil. Carefully place dough on hot grill. The dough will begin to puff almost immediately. When the bottom crust has lightly browned, turn the dough over using two spatulas. Working quickly, brush oil over crust, and then brush with 2 tablespoons tomato sauce.
Arrange 1/2 cup chopped tomatoes, 1/8 cup sliced black olives, 1/8 cup roasted red peppers over crust. Sprinkle with 1 cup cheese and 2 tablespoons basil. Close the lid, and cook until the cheese melts. Remove from grill, and set aside to cool for a few minutes while you prepare the second pizza.