Preheat your Traeger Grill to 225* with the super smoke feature enabled.
Unwrap the chicken quarters and pat them dry with a paper towel.
Season the chicken quarters all over with seasoning, including under the skin.
Place the seasoned chicken on the grill grates and close the lid.
Allow the chicken to smoke for at least 30 minutes-- and up to 1 hour-- for great flavor.
Preheat a large dutch oven to high heat and add a thin layer of oil to the pan.
Sear each piece of chicken on both sides until golden brown-- about 3 minutes per side.
Transfer the chicken to a plate and repeat with remaining pieces.
Once the chicken is seared, remove the pan from the heat and stir in the vegetables.
Cook for a minute or two and then stir in the tomato sauce and chicken broth.
Stir to combine and add the vinegar.
Then submerge the seared chicken pieces in the liquid, covering as much as possible.
Cover the pan with the lid and return it to the Traeger set to 275*.
Braise the chicken for about an hour and check for doneness.
Continue cooking until the chicken begins to fall off the bone (about 90 minutes total).
Allow the pan to cool briefly and then remove the chicken from the sauce.
Using forks or gloved hands, pull the chicken from the bones.
Discard the bones and skin and toss the meat with a bit of the reserved sauce.
Use this delicious Traeger braised chicken quarter meat in a variety of dishes!