Heat a large pot or Dutch oven over medium heat. Add 1 tablespoon vegetable oil and wait until it shimmers, about 30 seconds.
Add andouille sausage rounds in a single layer. Cook without stirring for 2 minutes, then flip and cook another 1-2 minutes until browned and crispy on the edges. Remove sausage to a plate and set aside. Do not clean the pot.
Add diced onion, celery, and red bell pepper to the pot with the sausage oil. Cook over medium heat for 5-8 minutes, stirring occasionally, until onions are translucent and vegetables are soft.
Add minced garlic and stir constantly for 1 minute until fragrant.
Add Cajun seasoning, salt, black pepper, paprika, and cayenne pepper. Stir for about 30 seconds to bloom the spices.
Pour in chicken broth and add cubed potatoes. Stir to combine. Increase heat and bring to a gentle boil, then immediately reduce to low heat.
Simmer uncovered for 20-25 minutes, until potatoes are fork-tender.
Return the browned sausage to the pot. Stir in heavy cream and shredded cheddar cheese. Stir gently until cheese melts completely.
Let simmer for 5 more minutes to allow flavors to marry and soup to thicken slightly. Taste and adjust salt and pepper as needed.
Serve hot, garnished with fresh chopped parsley.