Clean grill, greasing it lightly with olive oil. Heat the grill using gas or coals to a medium-high heat. Grills vary, so adjust temperature accordingly.
Place the dough onto the grill, and cook for 1 to 2 minutes on each side, or until golden brown. Be careful not to burn the crust, as it cooks very quickly.
Remove the crust from the grill and add the toppings. Spread pesto evening across the pizza, crumbling a layer of goat cheese over it. Cover this with thinly sliced tomato. Sprinkle arugula over tomatoes, and add a dusting of goat cheese.
After the toppings have been added, cover the pizza on the "warming zone" of your grill, and let the cheese soften. Goat cheese will not "melt" as others, such as mozzarella, will.