Cut broccoli into small florets. Peel tough stem exteriors; dice tender centers into 1/4-inch pieces.
Melt butter in a soup pot over medium. Cook leeks 7 minutes; add stems, beans, broth, salt, and pepper. Simmer partly covered 18 to 20 minutes until stems soften.
Meanwhile, boil water under a steamer basket. Add florets; cover and steam 4 to 6 minutes until just tender. Remove basket; discard water.
Remove soup from heat; blend smooth with an immersion blender. Add milk; warm gently 3 minutes.
Turn off heat; wait until bubbling stops. Stir in cheddar by small handfuls until melted; fold in florets.
Divide among 6 bowls; serve hot, keeping cheese soup below a simmer.