Step 1: Place the chicken breasts and thighs in a large pot or Dutch oven, ensuring they lie in a single layer if possible.
Step 2: Add the chicken broth, salt, pepper, garlic, onion, and bay leaf to the pot. The liquid should cover the chicken by at least an inch.
Step 3: Place the pot over medium-high heat and bring the liquid to a very gentle simmer. Do not let it reach a full, rolling boil.
Step 4: Reduce heat to low and cover the pot. Cook for 12-15 minutes, or until the thickest part of the chicken reaches an internal temperature of 165°F (74°C).
Step 5: Remove the chicken from the pot and let it rest on a cutting board for 5 minutes to allow the juices to redistribute.
Step 6: Use two forks to shred the meat, or place it in a stand mixer with a paddle attachment on low speed for 30 seconds.
Step 7: Toss the shredded chicken with a tablespoon or two of the remaining cooking liquid to keep it moist before serving or storing.