Make the dough: Put 1 1/2 cups warm water (110 degrees F to 115 degrees F) in the bowl of a stand mixer fitted with the dough hook attachment. Sprinkle the yeast over the water and let stand until foamy, about 10 minutes. Whisk the flour, sugar and 1 teaspoon salt in a separate bowl, then add to the yeast mixture along with 2 tablespoons olive oil. Mix on low speed until the dough starts to come together, about 1 minute. Increase the speed to medium low and mix until the dough starts to look smooth, about 1 minute. Increase the speed to medium and mix until the dough is smooth and elastic, 4 to 6 more minutes (it should stretch 3 inches without breaking). Transfer to an oiled bowl and cover with a damp towel or plastic wrap. Let rise in the refrigerator overnight.
Lightly press down on the dough, then divide into 6 balls. Put on a baking sheet, cover and refrigerate until ready to grill.
Meanwhile, make the toppings: Heat the olive oil in a pot over medium-low heat. Add the garlic and red pepper flakes and cook 3 to 5 minutes. Crush the tomatoes into the pot with your hands; add the juices. Rinse the can with 1 cup water and add to the pot. Add a handful of basil and season with salt and pepper. Increase the heat to medium and simmer until thickened, 20 to 25 minutes. Cool slightly.
Preheat one side of a grill to high and the other to medium. Brush 2 baking sheets with olive oil; stretch each dough ball into a thin oval (about 8 by 11 inches) on the sheets. Brush the tops of the dough with more oil.
Working in batches, grill the dough over high heat until bubbly on top and marked on the bottom, 3 to 4 minutes. Flip the dough onto the cooler side of the grill. Quickly top each crust with tomato sauce and mozzarella and cover until the cheese melts, about 2 minutes. Transfer the pizzas to a cutting board; sprinkle with the Parmesan and remaining basil.
Recipe courtesy of Joe and Jill Biden for Food Network Magazine