Combine arrowroot and water in a small bowl until smooth; set aside.
Melt butter in a medium saucepan. Add green onion and sauté until soft. Add milk and broth; bring to a boil. Reduce heat; add broccoli and carrots. Return to a simmer and cook over low heat for 5 minutes or until carrots are tender. Gradually stir shredded cheese into soup in batches. Stir until smooth.
Using an immersion blender, briefly blend soup until desired degree of smoothness. Whisk in arrowroot mixture, stirring constantly until thickened. Season with salt and pepper.