Heat the olive oil in a large stockpot over medium heat. Once the oil is glistening, add the onions and carrots and cook until the onions begin to soften, about 5 minutes. Add the garlic and cook, stirring, until fragrant, about 1 minute more.
Add the butter to the pot. Once the butter is bubbling, sprinkle the flour over the mixture and stir until a thick paste forms, about 1 minute. Slowly stir in the chicken stock, then bring to a boil over high heat. Reduce the heat to a simmer, add the salt, pepper, rice, and cabbage and cook, stirring, until the rice is tender, and the soup has thickened, about 20 minutes more.
Stir in the milk, dill, lemon juice, and chicken. Let simmer until the chicken is warmed through, about 5 minutes.
Ladle the soup into bowls and serve topped with dill and pepper.