Soak the mushrooms in 2/3 cup hot water, 10 minutes. Strain through a paper towel-lined sieve, reserving the liquid. Rinse the mushrooms and finely chop.
Meanwhile, toss the chicken and carrot with the olive oil in a 6- to 8-quart slow cooker; season with salt and pepper. Top with the chopped mushrooms, garlic and basil sprigs, then pour in the tomatoes and their juices. Whisk the red wine with the flour, tomato paste and reserved mushroom liquid in a bowl until smooth; add to the slow cooker. Cover and cook on low, 7 hours.
About 10 minutes before the chicken is done, cook the rice as the label directs. Uncover the slow cooker and stir, breaking up the chicken into chunks; let stand, uncovered, until the sauce is thickened, about 15 minutes. Discard the basil sprigs and season with salt and pepper. Serve the chicken with the rice; top with torn basil.