Pour the water into a large pan. Add the carrots and boil until tender-crisp, about 10 minutes. Drain the carrots and set aside in a serving bowl.
In a small saucepan over moderate heat, combine the apple juice and cornstarch. Stir until the mixture thickens, about 5 minutes. Stir in the mint and cinnamon.
Pour the mixture over the carrots. Serve immediately.