1. Separate broccoli stems from florets. Using a vegetable peeler, peel stems to remove tough outer layer, then slice into 1/4 inch thick "coins." Break or cut the florets into small pieces. Reserve stems and florets separately.
2. In a medium saucepan, heat oil and butter over medium heat. Add leaks and cook, stirring often, until softened and fragrant, about 3 minutes.
3. Add broccoli stems, potato, and garlic, and cook 2 to 3 minutes.
4. Add 3 cups of water, broth, salt, and pepper; bring to a boil.
5. Reduce heat; cover partially, and simmer until broccoli and potato are tender, about 12 minutes.
6. Add florets, bring to a boil and then simmer 5 minutes. Transfer soup in batches to a blender or food processor, and puree until smooth.
7. Return soup to saucepan; add half and half and chives, and reheat briefly.