Heat oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Stir in caraway seeds and star anise; cook, stirring occasionally, until toasted and aromatic, about 1 minute. Stir in onion; cook, stirring occasionally, until tender and golden, about 4 minutes. Stir in beets; cook, stirring occasionally, until slightly tender, about 3 minutes. Stir in cabbage, carrot and potato; cook, stirring occasionally, until the cabbage is slightly wilted, about 3 minutes. (Decrease heat to medium, if needed, to prevent burning.) Stir in broth and cider; adjust heat to maintain a simmer. Cover and cook, stirring occasionally, until the cabbage, potato and beets are softened, about 40 minutes.
Meanwhile, combine yogurt, lemon juice, 1 tablespoon dill, 1 teaspoon mint and ⅛ teaspoon salt in a small bowl. Cover and refrigerate until ready to use.
When the soup is finished cooking, stir in vinegar, pepper and the remaining 1 tablespoon dill, 2 teaspoons mint and ½ teaspoon salt. Remove and discard the star anise. Divide the soup among 6 bowls. Garnish with ground pepper, dill and/or mint, if desired. Top with a dollop of the yogurt cream.
