Heat oil in a large saucepan over medium heat. Add onion, ginger, and caraway seeds. Cook, stirring occasionally, until onions are soft, about 7 minutes. Stir in bell peppers and beet, and cook until peppers begin to soften, about 5 minutes. Add stock and 3 cups water; bring to a boil. Stir in salt; season with pepper. Reduce heat; simmer, partially covered, until peppers and beet are tender, about 45 minutes, stirring occasionally.
Stir in dill, cilantro, and lemon juice. Let cool completely. Refrigerate, covered, until cold, about 1 1/2 hours. Divide soup among 4 bowls; top each with 1 tablespoon yogurt. Garnish with dill sprigs.