Heat butter in medium saucepan over medium-high heat. When foaming subsides, add the grated carrots and ½ teaspoon ground cinnamon; cook, stirring frequently, until carrots soften, about 2 minutes. Add couscous and figs and cook, stirring frequently, until grains are just beginning to brown, about 5 minutes. Add water, broth, and salt and pepper; stir briefly to combine, cover, and remove pan from heat. Let stand until grains are tender, about 7 minutes.