Peel and thinly slice the red onion. Melt the butter in a large non-stick skillet over low heat. Add the onion and cook slowly, stirring often, for about 30-45 minutes, until nicely caramelized. Season with a pinch of salt and set aside.
Preheat the oven to 500 degrees (or as hot as it will get). Line a baking sheet with foil and spray with oil.
Place the pizza dough on the prepared baking sheet and gently press to fill the pan. If the dough won't stretch, let rest for a few minutes and try again.
Bake the pizza for 5-7 minutes in the preheated oven, until the top of the dough is dry but the pizza is not yet browned.
Use a brush to lightly cover the pizza dough with olive oil, brushing the oil all the way to the edges (this helps create the crispiest crust).
Evenly distribute the goat cheese, nectarine slices, and caramelized onions over the pizza crust, and then sprinkle with the mozzarella cheese.
Bake the pizza for 10 minutes or so, or until the crust is as browned as you like it. Remove from the oven and immediately top with arugula and season with freshly ground black pepper. Let the arugula wilt for a couple of minutes before slicing and serving.