1. Scrub potatoes and cut into 1/2-inch chunks.
2. In a 4- to 5-quart pan over high heat, combine potatoes, carrots, and broth. Cover, bring to a boil, and cook until potatoes and carrots are tender when pierced, 10 to 12 minutes.
3. Meanwhile, remove any bits of shell from crab.
4. Add buttermilk to pan and stir until hot, 2 to 3 minutes.
5. Add crab, mix gently, and ladle soup into wide bowls. Add cucumber equally to portions. Season to taste with salt and pepper.