To make the chive cream, in a small blender or food processor, combine the chopped chives, yogurt, buttermilk, sugar and salt. Process until well blended. Cover and refrigerate until needed.
In a large saucepan, heat the olive oil over medium heat. Add the onion and saute until soft and lightly golden, about 6 minutes. Stir in 1/4 teaspoon of the salt. Add the carrot and saute until the carrot is softened, about 5 minutes. Add the garlic and saute for 1 minute. Stir in the split peas, water, stock, the remaining 1/4 teaspoon salt and the pepper and bring to a boil. Reduce the heat to low, cover partially, and simmer until the peas are tender, 60 to 65 minutes.
In a blender or food processor, puree the soup in batches until smooth and return to the saucepan over medium heat. Reheat gently.
Ladle into warmed individual bowls. Top each serving with a drizzle or swirl of chive cream and garnish with a sprinkle of chopped ham and a few long cuts of fresh chives. Serve immediately.