For the dough, in a small bowl, combine the water, yeast, sugar and olive oil. Let stand until foamy, about 5 minutes.

In the bowl of a food processor, pulse both the flours and salt. Add the yeast mixture and process until a ball of dough forms. Transfer the dough to an oiled bowl. Cover and let rise until doubled in volume, about 2 hours.

Preheat a gas or charcoal grill on high heat.

On a lightly floured work surface, roll out the dough into an 8-inch by 12-inch rectangle.

Toss the vegetables in olive oil, salt and pepper.

Grill the corn, turning occasionally, until lightly charred, about 5-7 minutes. Grill the peppers, zucchini, onions and broccoli until grilled marked, about 3-4 minutes. Grill the tomatoes until their skin just starts to split, about 2 minutes.

Slice the kernels off the corn. Dice the peppers, zucchini, onions and broccoli. Halve the tomatoes.

Dicing the zucchini, squash and peppers and slicing the kernels off the corn cobs.

Place the rolled-out dough directly on the grates and grill for 3 minutes with the lid closed until the crust has grill marks and has puffed up. Turn over the crust and grill for an additional 2 minutes.

Remove the crust from the grill. Spread the tomato sauce leaving a 1/2-inch border at the edge and scatter the cheese on top. Arrange the vegetables in stripes by color. Return the pizza back on the grill for a couple minutes to let the cheese melt.