In a large saucepan, heat oil over medium heat. Add onion, garlic, ginger, thyme, salt, and pepper. Cook, stirring occasionally, until softened, about 7 minutes.
Meanwhile, pierce 2 medium, unpeeled sweet potatoes all over with a fork. Cook in microwave until tender, about 10 minutes. Cool briefly, peel and add flesh to the pan. Add broth and milk. Bring to a simmer, stirring occasionally, then simmer 5 minutes. Transfer soup in batches to blender and puree until smooth. Return to the pan and heat through. Adjust seasonings and serve hot.