In a large Dutch oven or heavy-bottomed pot, melt the butter over medium heat.
Add diced onions, carrots, and celery. Season with salt and pepper. Sauté for 5–8 minutes until softened. Add minced garlic and stir for 1 minute.
Sprinkle flour over the sautéed veggies and stir to coat evenly, cooking for 1–2 minutes.
Add the chicken breasts (or thighs), potatoes, red bell pepper, thyme sprigs, and bay leaf. Pour in chicken broth and season with additional salt and pepper.
Bring to a gentle simmer. Cover and cook for 25–30 minutes, or until chicken is cooked through and vegetables are tender.
Add green beans during the last 10 minutes of simmering.
Remove bay leaf and thyme stems. Shred the chicken with two forks and return it to the pot. Stir well to combine.
Serve hot, garnished with fresh thyme or parsley if desired.