Preheat oven to 425°.
Combine first 4 ingredients in a medium bowl, stirring with a whisk. Add figs; toss to coat. Remove figs with a slotted spoon, reserving vinegar mixture.
Place figs in a cast-iron or ovenproof skillet coated with cooking spray. Bake at 425° for 8 to 10 minutes. Remove figs from pan; place on a plate. Immediately add reserved vinegar mixture to hot pan, scraping pan to loosen browned bits. Pour into a small bowl; let figs and vinaigrette cool to room temperature.
Place arugula on a platter; arrange figs over arugula. Sprinkle with cheese and pepper. Drizzle with cooled vinaigrette.