Grill the sausage, turning, until cooked through. Cut into slices. (Alternatively, remove the sausage from its casings, cook on a griddle on the grill and crumble.)
Meanwhile, put the Canadian bacon, onion slices and mushroom cap on the grill. Grill for 2 to 3 minutes, then flip and grill until the onion and mushroom are slightly charred and tender, another 2 to 3 minutes. Let cool for a few minutes, then roughly chop.
To set up the pizza stone on the grill, stack 2 bricks on top of each other on one side of the grill and 2 more bricks on top of each other on the other side of the grill, about 12 inches apart to create a platform for the pizza stone. Put the pizza stone on top of the bricks to elevate it over the grates. The stone should be 5 to 6 inches above the grates. Raise the heat to medium high, cover and preheat for 30 minutes to allow the pizza stone to get nice and hot.
On a floured work surface, press the pizza dough into a round 12 to 14 inches in diameter and about 1/4 inch thick. Dust a pizza peel or the back of a baking sheet with the cornmeal. Transfer the dough to the pizza peel or baking sheet.
Top the dough with the sauce and spread it out with a spoon into a thin layer. Top with the pepperoni, sausage, Canadian bacon, onions, mushroom and mozzarella. Carefully slide the pizza onto the pizza stone. Cover the grill and cook until the crust is golden brown and the cheese is melted, 12 to 14 minutes. Garnish with a drizzle of olive oil and the basil. Slice and serve hot.