Place beets into a pot, cover with water, and add vinegar. Bring to a boil. Reduce heat to medium-low and simmer until halfway tender, about 30 minutes. Drain. Let cool until safe to handle, at least 15 minutes.
Place beets on a cutting board. Remove skin and chop beets.
Heat oil in a large pot over medium heat. Cook and stir potatoes, onion, red chile flakes, and curry until potatoes are slightly softened, about 15 minutes. Add the boiled beets, chicken stock, and brown sugar. Increase heat to high; bring soup to a boil. Reduce heat to low and cover; cook until vegetables are tender, about 30 minutes. Season with salt.
Fill blender halfway with soup. Cover and hold lid down with a potholder; pulse a few times before leaving on to puree. Pour into a pot. Repeat with remaining soup.