In a large pot, combine onions, garlic, carrots, chicken and thyme. Season with salt and pepper. Add cold water to cover. Bring to boil then reduce heat to a simmer. Skim and discard any scum that may come to the top.
Simmer until the meat of the chicken falls off the bone with almost no pressure from a fork, about 1 to 1 1/2 hours. Remove the chicken pieces to a plate. Use 2 forks to separate the meat from the bone. Add the meat back to pot.
Add the potatoes, cook until fork tender, about 20 minutes. Add green beans, cook until crisp-tender, about 3 to 4 minutes. Remove from heat, add hot sauce, salt and pepper to taste.