Heat oil in a Dutch oven over medium heat, and add andouille sausage. Cook until browned, 1 to 2 minutes on each side. Remove from the pan and set aside.
Add onions, green bell pepper, red bell pepper, celery, and garlic to the Dutch oven. Sprinkle with half the Cajun seasoning; stir, and allow to cook for 3 minutes.
Add chicken broth and potatoes, stir and bring to a boil; reduce heat and simmer until potatoes begin to soften, about 5 to 10 minutes.
Carefully take 3 cups of the potato mixture from the Dutch oven, and put into a food processor. Process until smooth; add up to 1/2 cup broth as needed. Pour back into the Dutch oven and stir.
Add andouille sausage, crawfish tails, heavy cream, and remaining Cajun seasoning. Stir to combine well, turn heat to low, cover, and allow flavors to blend for 5 minutes.
