Using a food processor, soak the dried mushrooms in 1 cup chicken broth for 10 minutes. Process until coarsely chopped, about 20 seconds.
In a large pot, heat the olive oil over medium-high heat. Add the carrots, onion and sage and cook, stirring, until crisp-tender, about 5 minutes. Add the rehydrated mushroom mixture and cook, stirring, for 1 minute; season with salt and pepper.
Stir in the remaining 5 cups broth, the rice mix and the sliced mushrooms and bring to a boil. Add the chicken breast, lower the heat and simmer, stirring occasionally, until the rice is tender and the chicken breast is cooked through, about 25 minutes. Transfer the chicken to a plate and shred; return to the pot. Stir in the heavy cream; season with salt and pepper and simmer for about 5 minutes.