Preheat oven to 400. Combine sugar, cayenne, and teaspoon salt in a small bowl.
Heat oil in a large nonstick skillet over medium heat. Add onion, and cook, stirring occasionally, until light gold, about 12 minutes. Add 1/3 of the figs, and cook until soft, about 5 minutes. Add 1 tablespoon sugar mixture, and cook, stirring often, until sugar caramelizes, about 10 minutes. Let cool slightly, then pulse onion-fig mixture and walnuts in a food processor. Add pepper and remaining 1/2 teaspoon salt. (Spread can be refrigerated for up to 2 days.)
Place remaining figs, cut sides up, on a rimmed baking sheet. Sprinkle with remaining sugar mixture. Roast until tender and caramelized on cut sides, about 10 minutes.
Spread 3 tablespoons onion-fig mixture on each slice of bread. Top each with arugula, cheese, and several warm fig halves, and cut into 3 or 4 pieces. Serve immediately.