Heat butter and olive oil in a large pot over medium heat. Add onions, carrots, and celery; cook until softened, about 5 minutes.
Add salt, pepper, garlic, and ginger; cook for 1 minute. Pour in chicken broth, add chicken breasts, and bring to a boil. Reduce heat, cover, and simmer for 15 minutes.
Remove chicken, shred with forks, and return to the pot. Heat through for 3-5 minutes, then stir in dill and lemon juice. Season with additional salt if needed.
Notes
For extra flavor, garnish with additional dill or lemon slices before serving.