Prepare a grill for medium heat.
Roll out the pizza dough and form into a circle. Put the dough on the grill, cook for 2 minutes and then flip with tongs. Move to a cooler spot on the grill and cook until the dough begins to pull away from the grill, about 2 minutes.
Remove the dough to a work surface. Add the sauce, cheese, pepperoni, sausage, Caramelized Onions and mushrooms and place back on the grill over indirect heat. Cook until the cheese melts, about 5 minutes. Garnish with basil and serve warm.
Melt the butter in a medium skillet over medium heat. Add the onions and a pinch of sugar if using and toss to coat the onions. Cover and cook, stirring occasionally, until the onions turn golden, about 10 minutes. Turn the heat up to medium high and continue to cook, stirring constantly, until the onions begin to brown, another 10 minutes. Season with salt and pepper.