Combine carrots and turnips in a Dutch oven. Add 4 cups broth. Bring to a boil; cook 10 minutes. Add broccoli and asparagus; cook 5 minutes. Drain.
Whisk together remaining 2 cups broth and cornstarch. Cook in a large skillet over medium-high heat 5 minutes or until slightly thickened. Stir in pepper. Add arugula; cook just until arugula wilts. Drizzle over cooked vegetables.
*2 cups chopped fresh spinach may be substituted.