Place the onion and chicken in a pot with enough cold water to cover. Season with salt, and bring to a boil. Cook 30 minutes, or until the chicken meat is easily removed from the bone. Remove chicken from the pot, reserving water. Discard the onion. Pull all the meat from the bones, chop, and return to pot. Discard bones.
Place the cauliflower, Brussels sprouts, baby carrots, and asparagus in the pot. Pour in the chicken broth. Season with garlic powder and salt-free seasoning blend. Bring to a boil, reduce heat to low, and simmer 40 minutes.
Stir the rice into the pot. Continue cooking 20 minutes, or until rice is tender. Mix dill into the soup 5 minutes before serving.