Heat oil in a large pot over medium heat. Stir in onion, bell pepper, carrot, and garlic; cook until tender, about 5 minutes. Mix in 1/2 of the black beans, then add chicken broth.

In a blender, purée remaining beans until smooth; stir into the pot and bring soup to a boil. Reduce the heat to low. Mix in ham, salt, cumin, and pepper; simmer for 20 minutes. Garnish with Cheddar cheese and sour cream to serve.