Preheat oven to 450°F. Heat oil in a saucepan over medium. Add red onion and 1/4 teaspoon each of the kosher salt and pepper. Cook, stirring, until tender, 2 minutes.
Roll pizza dough into a 1/4-inch-thick, 17- x 11-inch rectangle. Place on a large baking sheet lined with parchment paper. Sprinkle with remaining 1/4 teaspoon each salt and pepper. Top evenly with cooked onions, one-third of the fig slices, and cheese. Bake until crust is lightly browned and done, 15 to 20 minutes.
Meanwhile, cook balsamic vinegar in a small saucepan over medium-high until reduced by half, about 5 minutes. Top flatbread with remaining figs, arugula, and balsamic reduction.