Melt butter in a large saucepan over medium-high heat. Add leeks and celery, stirring occasionally, and cook until softened, 6-8 minutes.
Add garlic and cook, stirring constantly, for about 1 minute, or until fragrant.
Add broccoli and broth, then bring to a boil. Reduce heat to medium, cover, and cook for 12 minutes or until the broccoli is tender.
Stir in thyme and salt for a fresh, herbaceous flavor boost.
Puree the soup with an immersion blender until smooth, about 2 minutes. (If using a traditional blender, transfer the soup in batches, taking care to remove the lid's center piece to let steam escape.)
Stir in the half-and-half and blend just until incorporated, about 15 seconds.
Serve hot, topped with the fresh, vibrant touch of chives.