Preheat oven to 425°. Spread butter over bottom of a small cast-iron skillet or shallow 1-qt. baking dish; sprinkle evenly with sugar.
Peel, quarter and core apples; arrange in a circular pattern over sugar, rounded side down. Place figs around apples. Place skillet over medium heat; cook until sugar is caramelized and apples have softened slightly, 10-12 minutes. Remove from heat; top with cheese.
Unroll crust; place over fruit and cheese, tucking under edges. Place skillet in oven on an upper rack; bake until crust is golden brown, 15-18 minutes. Let cool in skillet 5 minutes. Carefully invert onto a serving plate; serve warm.