Warm the oil in a large pot over medium heat.
Soften the onion, carrots, and celery with a pinch of salt, stirring often, 6–8 minutes.
Stir in the garlic and tomato paste; cook until fragrant, about 1 minute.
Add the cabbage and toss until it starts to wilt, 3–4 minutes.
Pour in broth and crushed tomatoes. Add paprika, brown sugar, vinegar, pepper, and remaining salt; stir well.
Bring to a boil, then reduce heat and simmer gently until cabbage is tender, 25–30 minutes.
Finish with lemon juice and dill. Taste and adjust sweetness, tang, and salt.
Ladle into bowls and serve hot.