Make a dark roux with the flour and oil in a small, heavy pan, stirring constantly. The quantities will make about two or three times as much roux as you need for this recipe, but if you make any less it won't come out right. When the roux reaches the color of an old penny, remove it from the heat and stir in about 1/4 cup of stock until smooth. Let it cool. Reserve all but about 1/4 cup of the roux mixture for other uses.
In the same pan (after removing the roux to cool), heat the butter until bubbling and add the chopped bell pepper, onions, celery, and garlic. Cook until they're soft and the onions are clear.
Add the remaining stock to the pan and bring to a simmer. Whisk in 1/3 cup of the roux mixture until smooth. Add the rest of the etouffee base ingredients and cook on a low boil to reduce by about a third.
Slice the andouille sausages from end to end twice, to get four sticks from each sausage. Cut across the sticks to make quarter-round pieces.
In a heavy skillet over medium heat, cook the andouille to render out some of the fat and brown the edges of the andouille pieces noticeably. Pour off any excess fat. Add the duck, chicken, or turkey meat and the mushrooms to the pan and cook until the mushrooms begin to get limp.
Stir in one cup of the etouffee base. Lower the heat to medium-low. Cover the pan and cook for two or three minutes, until everything is piping hot and well imbued with the etouffee base.
Check the seasonings and add salty, pepper or Tabasco as needed. Serve over rice or on its own. Sprinkle green onions liberally over each plate and serve. Baked sweet potatoes would be the perfect side dish.
