How to Make It
In a small bowl, combine the yeast with the warm water and sugar. Let stand until foamy, about 5 minutes. In the bowl of a food processor, combine the all-purpose and whole wheat flours with the salt and process to blend. Add the olive oil and the yeast mixture and process until the dough comes together.
On a lightly floured work surface, knead the dough until smooth and elastic. Put the dough in a large oiled bowl and turn to coat. Cover and let rise until doubled in bulk, about 1 hour. Punch down the dough and cut it into quarters. Shape each piece of dough into a 6-inch disk.
Generously oil 2 large rimmed baking sheets and transfer 2 of the disks to each sheet; lightly brush the dough with olive oil. Let rest for 30 minutes.
Light a charcoal grill. Roast the bell peppers over a hot fire until charred all over. Transfer to a bowl, cover with plastic wrap and let steam for 10 minutes. Peel the peppers; discard the stems, seeds and cores. Cut the peppers into thin strips.
Rake three-fourths of the coals to one side of the grill, leaving a thin layer of coals on the other side. Working on the baking sheet, pat and stretch 1 dough disk to form a 9-inch round. Lift the dough up by the edge and drape it onto the grate over the hot part of the grill. Grill the dough until it is nicely toasted on the bottom, about 1 minute. Then rotate and cook for an additional minute.
Flip the crust over onto the cooler part of the grill and brush with olive oil. Sprinkle one-quarter of the cheese over the crust and top with one-quarter of the peppers, tomatoes, garlic and capers. Move the pizza back to the hot area of the grill to melt the cheese. Cook until the bottom of the crust is lightly charred, about 2 minutes; if it starts to burn, move it to the cooler side. Sprinkle with salt and pepper and serve immediately. Repeat with the remaining dough and toppings to make 3 more pizzas.