Cut broccoli into small florets. Peel the tough outer layer from stems and dice the tender centers into 1/4-inch pieces.
Melt butter in a soup pot over medium heat. Cook leeks for 7 minutes, then add diced stems, beans, broth, salt and pepper. Cover partly and simmer for 18 to 20 minutes until stems are soft.
Meanwhile, bring steaming water to a boil under a steamer basket. Add florets, cover and steam for 4 to 6 minutes until just tender; remove the basket and discard steaming water.
Remove the soup pot from the heat and blend the leek, stem and bean mixture until smooth with an immersion blender. Add milk and warm gently for 3 minutes.
Turn off the heat and wait until bubbling stops. Stir in grated cheddar in small handfuls until melted, then fold in steamed florets.
Divide among 6 bowls and serve while hot, keeping the finished cheese soup below a simmer.