Heat olive oil in a large nonstick skillet over low heat.

Add chopped carrots and finely chopped onion to the skillet.

Cover the skillet and cook over medium-low heat until the vegetables are just tender, stirring occasionally, about 15 minutes.

Add balsamic vinegar to the vegetables, cover, and cook until the vegetables are very tender and the vinegar has reduced, about 10 minutes longer.

Working in batches, add sliced beets with their liquid, honey, and the carrot mixture to a blender.

Puree the mixture until smooth.

Transfer the pureed mixture to a large bowl.

Mix in the buttermilk and season with salt and pepper to taste.

Chill the soup in the refrigerator until cold, about 3 hours.

To serve, ladle the chilled beet soup into bowls or mugs.

Top with chopped chives and sour cream (optional).