Sear the sausage: Heat the olive oil in a skillet over medium heat. Add the sliced andouille sausage and cook until both sides are golden brown, approximately 4-5 minutes. If needed, drain any excess fat.
Prepare the slow cooker: Transfer the browned sausage to the slow cooker. Layer the diced potatoes, onion, bell pepper, celery, and carrots over the sausage. Add the minced garlic, Cajun seasoning, salt, and black pepper.
Incorporate the broth: Pour the chicken broth evenly over the mixture. Stir gently to combine the ingredients thoroughly.
Cook: Cover the slow cooker and set it to cook on high for 3 hours or low for 5 hours, until the potatoes and vegetables reach a tender consistency.
Add milk: Once the primary cooking time has elapsed, stir in the milk and allow the soup to continue cooking on low heat for an additional hour.
Finish with cream and cheese: Stir in the whipping cream and shredded cheddar cheese. Allow the cheese to melt completely into the soup, which should take approximately 15 minutes.
Serve: Ladle the soup into bowls, then garnish with sliced green onions. Serve with a dash of Tabasco sauce for added spice, if desired.
