Melt the butter in a large Dutch oven or heavy-bottomed pot over medium heat. Add the onion, crystallized ginger, fresh ginger, garlic, sugar, and a pinch of salt. Cook, stirring occasionally, until the onions are soft and translucent, about 5 minutes.

Stir in the carrots, water, carrot juice, thyme sprigs, and baking soda. Increase the heat to high and bring the liquid to a boil, then reduce the heat to medium-low to maintain a steady simmer. Cook until the carrots are completely tender and break apart easily when pressed with a spoon, about 20 minutes.

Remove and discard the thyme sprigs. Working in batches, process the soup in a blender until velvet-smooth—fill the jar only halfway and hold the lid down firmly with a folded kitchen towel to keep the steam from forcing the lid off.

Return the pureed soup to the pot and stir in the vinegar. Adjust the seasoning with salt and pepper to taste. Serve hot with a dollop of sour cream and a sprinkle of chives.