Brown the Sausage: Heat vegetable oil in a large pot over medium heat. Add the sliced andouille sausage and cook for about 3-4 minutes, until browned. Once done, set it aside to keep the flavors intact.
Sauté the Vegetables: In the same pot, add diced onion, celery, and red bell pepper. Sauté for 5-8 minutes until they soften, then stir in minced garlic for another minute to infuse that delicious aroma.
Prepare the Soup Base: Now it's time to add flavor! Toss in Cajun seasoning, kosher salt, black pepper, paprika, and cayenne pepper. Pour in the chicken broth along with the cubed potatoes. Simmer on low for 20-25 minutes, or until the potatoes are fork-tender and bursting with flavor.
Finish the Soup: Return the browned sausage to the pot. Add heavy whipping cream and shredded cheddar cheese, stirring until the cheese melts and the soup is creamy, about 5 minutes more.
Garnish and Serve: Ladle the soup into bowls and garnish with fresh parsley. Enjoy it warm alongside some crusty bread or your favorite crackers.