Butter a 2 1/2- to 3-quart baking dish with the butter and set aside.
Heat the oil in a large, heavy skillet over medium-high heat and saute the andouille until beginning to brown on all sides, about 4 minutes. Add the onions and celery and cook, stirring often, for 4 minutes, until soft. Add the garlic and cook for an additional minute. Remove from the heat and set aside.
In a large bowl combine the eggs, cream, milk, salt, cayenne and black pepper, and mix well. Add the bread, andouille mixture and half of the cheese. Fold the ingredients to mix thoroughly. Cover and refrigerate for 30 minutes.
Preheat the oven to 375 degrees F.
Transfer the mixture to the prepared baking dish and sprinkle with the remaining cheese. Bake in the preheated oven for 45 minutes, or until golden brown on top.
Remove pudding from the oven and let rest for 5 minutes before serving.
Yield: 8 - 10 servings