If using a whole chicken remove the breasts, wings, thighs, and legs. Save those for another recipe. Place the wings, and remaining chicken bones on the lined sheet pan and roast for 30 minutes. If using a rotisserie chicken place the bones on a lined sheet and roast.
While the bones are roasting in the stock pot sauté the onion, celery, carrots, and garlic in the oil until the vegetables start browning. Stir infrequently until all of the vegetables have browned. If the whole chicken came with additional parts add them to the vegetables and cook until browned as well.
Reduce the heat add the water, oregano, thyme, half of the salt, and the pepper. Cover and let simmer. Once the bones and wings are done add them to the pot and simmer for another 4 hours. Make sure that the bones are completely submerged in water.
Strain the bones and vegetables out. Season with the rest of the salt to taste and then cool immediately or use in another recipe.