Prepare a gas or charcoal grill to high heat. Brush the grill with a bit of oil.
Shape the dough into 2 medium-ish pizzas while on a clean floured surface. Let the dough sit for 5 minutes and then re-form to make sure it's as big as you'd like. Place the pizza dough on a lightly floured rimless baking sheet, or pizza peel.
Transfer the dough over to the grill, and slide the pizza dough directly onto the grill. Grill the dough for about 2 minutes with the lid closed. Using a pair of tongs, lift the lid, and flip the dough over onto the other side and cook for about 2 minutes more. The dough should be golden brown with grill marks on both sides.
Once the dough is cooked, remove it from the grill and place it back onto the baking sheet.
Brush the olive oil over the pizza and sprinkle with garlic. Arrange the mozzarella, red onions, goat cheese and mushrooms on top and place back onto the grill until the cheese has melted.
Using tongs, remove the pizza from the grill and season with salt and pepper as needed. Garnish with long strips of parmesan cheese and basil leaves. Slice and serve immediately.