Combine spices in a small bowl and store in a cool, dark place in a tightly sealed container.
Pick all the meat from the chicken (discard skin) and use your hands to shred it into pieces, or chop into medium pieces, as you prefer. Save all the juice and fat from the roasting pan (or container) as well and set aside; refrigerate chicken until needed.
Place the carcass, quartered onion, and vegetable trimmings into a large pot to make broth. Add 10 cups of water, bring to a boil, reduce heat and simmer for about 1 hour. Strain the broth and discard solids. You should have about 6 cups.
Heat the oil in a medium cast-iron skillet over medium-high heat and add the sausage. Sear until the sausage starts to color. Parts of the sausage will begin to stick to the pan. When there is a good coating stuck to the pan pour in 1/4 cup chicken broth and scrape it loose. Let this cook until all the liquid has evaporated.
Transfer the sausage to a plate and set aside.
Return the pan to the heat and add the butter. When it melts, add the diced onion and cook about 10 minutes, until a nice deep brown color. About halfway through the onion should start to stick to the pan; deglaze with 1/4 cup chicken broth and let this reduce until the skillet is dry. When the onion starts to stick again, add 1/2 cup broth; when this is almost gone, add the bell peppers, jalapenos, scallions, celery, garlic, spice mix, salt, bay leaves, oregano, and tomato paste. Cook the vegetables for 10 minutes, stirring often, until they start to stick to the skillet. Deglaze with another 1/4 cup broth and reduce again until dry, then add the shredded chicken, 1 cup broth, and the juices from the chicken and reduce again by half.
Transfer the vegetable mixture to a heavy-bottomed pot and add the rice and the remaining 4 cups broth. You want this mixture to have plenty of room so the rice will cook more evenly. Cook, covered, over low heat for 40 minutes.
Remove pot from the heat and keep covered for 10 minutes more. If the rice seems unevenly cooked, leave the lid on a little longer and it will even out.
When jambalaya is done, transfer to a casserole dish and serve. (If left in the pot, it will overcook.).