Heat the olive oil in a large Dutch oven over medium-high heat. Add the turkey and cook, breaking it apart, until no longer pink, about 6 minutes.
Add the onion and garlic and cook until softened, about 4 minutes.
Stir in the smoked paprika, oregano, salt and pepper and cook for 30 seconds.
Add the cabbage and cook, stirring occasionally, until slightly wilted, about 5 minutes.
Add the crushed tomatoes and broth. Bring to a boil, then reduce the heat, cover and simmer until the cabbage is tender, about 20 minutes.
Stir in the cooked brown rice and simmer 5 minutes more. Taste, adjust seasoning and serve.