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Melt butter in large pot over medium heat.
Add leeks and broccoli stems and cook until just beginning to soften, about 8 minutes.
Add garlic and cook until fragrant, about 1 minute.
Stir in broth and water and bring to boil.
Reduce heat to medium-low and simmer, covered, until stems are soft, about 8 minutes.
Add broccoli florets, cover, and simmer until tender, about 5 minutes.
Transfer soup to blender and puree until completely smooth, about 30 seconds.
Add mustard, milk, and cheese and puree until cheese is melted.
Season with salt and pepper.
Serve. (Soup can be refrigerated for up to 3 days. Reheat over medium-low heat; do not boil or cheese will separate.)