In a large pot, heat the vegetable oil over medium heat. Add the sliced andouille sausage and cook, stirring occasionally, until the sausage is well-browned on both sides. This should take about 3-4 minutes. Once browned, remove the sausage from the pot and set it aside on a plate or in a bowl.

Using the same pot, add the diced onion, celery, and red bell pepper. Cook over medium heat until the vegetables are softened, which typically takes about 5-8 minutes. Add the minced garlic and cook for an additional minute, being careful not to let the garlic burn.

Return the browned sausage to the pot along with the Cajun seasoning, kosher salt, black pepper, paprika, and cayenne pepper. Stir to combine.

Add the cubed potatoes and chicken broth to the pot. Increase the heat to bring the mixture to a simmer, then reduce the heat to maintain a gentle simmer. Cook for 20-25 minutes, or until the potatoes are fork-tender.

Once the potatoes are tender, stir in the heavy whipping cream and shredded cheddar cheese. Continue to cook for an additional 5 minutes, or until the cheese is melted and the soup is heated through.

Taste the soup and adjust seasoning as needed. If you prefer a thicker soup, you can mash some of the potatoes directly in the pot or use an immersion blender to blend a portion of the soup.

Ladle the soup into bowls and serve warm. If desired, garnish with additional shredded cheese, a sprinkle of Cajun seasoning, or chopped parsley for a touch of color.