Remove the dough from the refrigerator, if necessary, about 1 hour before grilling so it will be easier to shape. To easily cut the rind from the Brie and cut the cheese into chunks, place it in the freezer for 30 to 45 minutes until semi-firm before cutting.
In a large skillet over medium heat, warm 2 tablespoons oil. Add the onion and 1 tablespoon water. Cook for 5 minutes, stirring often. Reduce the heat to medium-low and continue cooking until the onion is softened and golden brown, about 20 minutes, stirring frequently and occasionally adding more water, a tablespoon at a time, to prevent scorching. Stir in the rosemary, remove from the heat, and let cool.
Prepare the grill for direct cooking over medium heat (350° to 450°F) and preheat a pizza stone, following manufacturer's instructions.
In a small bowl whisk 2 tablespoons oil with the salt and ¼ teaspoon pepper.
Divide the dough into two equal balls. On a lightly floured work surface, gently press, stretch, and pull each dough ball into an oval or rectangle about 12 inches at its longest point, about 6 inches at its widest point, and about ⅓ inch thick; don’t worry if the shape is irregular. (If the dough retracts, cover it with a kitchen towel, let it rest for 5 minutes, and then continue.)
Transfer your first piece of dough onto a pizza peel (or a rimless baking sheet) lightly coated with flour. Brush the dough with half of the seasoned oil, and then distribute half the onions, half the cheese, and half the capers, leaving a ½-inch border. Slide the first pizza onto the preheated pizza stone and cook over direct medium heat, with the lid closed, until the cheese is melted and the crust is golden brown, 11 to 13 minutes, occasionally rotating the pizza for even cooking.
Using a pizza peel or a large spatula, remove the pizza from the stone and immediately top with half of the arugula. Drizzle lightly with oil and season with pepper, if desired. Cut crosswise into slices and serve immediately.
Repeat steps 6 and 7 with the remaining dough and toppings.