Place the chicken in a large Dutch oven, or heavy duty pot.
Pour enough cold water to cover by 1 inch.
Add the roughly cut carrots, onions, celery, and the bay leaf.
Bring to a boil over high heat.
Reduce the heat to medium, partially cover, and simmer until the chicken is tender, and almost falling of the bone....about 2 hours.
Turn off the heat and set aside to cool.
When the chicken is cool enough to handle, remove the skin, and then the meat - shred the meat into large bite-size pieces.
Strain the broth into a large saucepan, or stock pan, and wipe the Dutch oven (or large pot) clean.
Bring the broth to a simmer.
Meanwhile, melt the butter over medium heat in the Dutch oven, or pot.
Add the remaining carrots, onion and celery. Cook until tender, about 6 minutes.
Add the flour and whisk until smooth.
Constantly stir the roux for about another 5 minutes.
Add the heated chicken broth. Stir until slightly thickened, about 5 minutes.
Season with salt and pepper. Taste and add more salt, if desired.