Place chicken, onion, tomato, carrot, and 2 cloves of garlic into a large pot. Add water to cover, then bring to a boil. Skim and discard the foam that has floated to the surface. Reduce heat to medium-low, and simmer uncovered until the chicken is tender, 30 to 40 minutes. Place the chicken into a bowl and shred it into strips as thick as a pencil; set aside. Strain the cooking liquid and discard the vegetables.
Heat the olive oil in a large skillet over medium heat. Stir in the garlic, onions, green peppers, and red peppers. Cook and stir until the vegetables have softened, 3 to 4 minutes. Stir in tomato puree, white wine, chicken, and enough of the reserved broth to create a thick sauce. Season with cumin, salt, and pepper. Simmer for about 5 minutes more to reduce the sauce slightly and coat the shredded chicken.
The chicken breasts may be cooked in a slow cooker, covered, on High for 3 to 4 hours.