Step 1Place the turkey carcass, meatiest side down, in a large stockpot. Fill the pot halfway with water. Bring to a boil and skim any scum off the surface. Reduce the heat and simmer 30 minutes. Remove the carcass to a carving board; let it sit until cool enough to handle.
Step 2Add the onion, carrots, celery, parsley, peppercorns and bay leaf to the water in the stockpot and continue simmering.
Step 3Shred or cut any remaining meat off the carcass and set it aside in a bowl; cover and refrigerate.
Step 4Return the bones to the stockpot and continue simmering, adding just enough water every hour to cover. Cook until the stock is rich and aromatic, at least 3 hours or up to 6.
Step 5Cool, then strain, discarding solids. (If you have time, refrigerate the stock until it's well-chilled, then lift off the fatty top layer and discard.)
Step 1Heat the oil in a Dutch oven or other large, heavy pot with a tight-fitting lid over medium heat until bubbly. Add the flour all at once and cook, whisking constantly, until the mixture turns deep mahogany brown, 25 to 30 minutes. Do not allow it to burn or you will have to start over.
Step 2When the roux is sufficiently colored, add the garlic, onion, celery, red and green pepper, thyme, cayenne, allspice and bay leaf. Stir quickly until the roux cools slightly and the vegetables start to soften. Stir in the salt. Cook, stirring frequently, until the vegetables are well-wilted, about 20 minutes.
Step 3If the stock has cooled, heat it in a saucepan. Raise the heat under the Dutch oven and pour in the hot stock. Bring the mixture to a boil while stirring rapidly.
Step 4Add the reserved turkey bits and the diced andouille. Reduce the heat and simmer until the flavors blend and the liquid is somewhat thickened, about 30 minutes. Stir in the artichoke hearts and simmer 5 minutes longer.
Step 5Season with pepper to taste and more salt if needed. Serve hot over rice, with chopped green onions for garnish and hot sauce for extra heat, or cool and refrigerate and serve the following day (the flavor only improves). Or cool the gumbo completely and freeze for up to six months.