Combine carrot, chicken, and garlic in an electric slow cooker, and add wine. Sprinkle with thyme, salt, and black pepper. Cover with lid; cook on low-heat setting for 8 hours.
Remove carrot, chicken, and garlic with a slotted spoon, reserving cooking liquid. Place 1/3 cup carrot, 3 garlic cloves, and 2 chicken thighs in each of 4 shallow bowls. Spoon 2 tablespoons of reserved cooking liquid over each serving.