Preheat oven to 400 degrees F.
Coat the sweet potatoes with 1 tablespoon of the oil and place in a small baking dish. Place the baking dish in the oven and roast the sweet potatoes for 50 minutes to 1 hour, or until the potatoes are fully roasted and fork tender. Once the potatoes are cool enough to handle, remove and discard the skin from the potatoes. Reserve the sweet potato flesh until ready to use.
Heat a 1 gallon stockpot over medium-high heat and add the remaining 2 tablespoons of oil. Add the andouille to the pot and saute until most of the fat has rendered and the meat is well caramelized, stirring constantly, 5 to 6 minutes. Use a slotted spoon to remove the andouille and drain on a paper-lined plate. Set aside until ready to use. Add the onions and saute until softened, 3 to 4 minutes. Add the minced garlic to the pot and cook, stirring, for 30 seconds. Add the cayenne pepper and chicken stock to the pot and bring to a boil. Reduce the heat to a simmer.
Add the roasted sweet potatoes to the soup and stir to blend. Simmer for 10 minutes, then blend with an immersion blender or in batches in a bar blender until smooth. Return the soup to the pot, if necessary, and stir in the heavy cream. Season with the salt and if necessary more pepper. Garnish with the cooked andouille, chopped chives and serve.