In a large nonstick skillet, heat oil over low heat. Add carrots and onion. Cover; cook over medium low heat until vegetables are just tender, stirring occasionally, about 15 minutes. Add vinegar, cover and cook until vegetables are very tender, about 10 minutes longer, or until the vinegar has reduced. Working in batches, add beets with their liquid, honey, and carrot mixture to blender. Puree. Transfer to large bowl. Mix in buttermilk. Season with salt and pepper. Chill until cold, about 3 hours. (Can be made 2 days ahead, covered and chilled.)
To serve, ladle into bowls or mugs. Top with chives and sour cream.