In a large pot or Dutch oven, heat the butter and olive oil over medium heat.

Add the onion, celery, and bell pepper. Cook for 5 to 6 minutes, stirring occasionally, until softened.

Stir in the garlic and smoked sausage and cook for 2 to 3 minutes until fragrant and lightly browned.

Sprinkle the flour over the mixture and stir constantly for 1 minute to form a light roux.

Slowly pour in the chicken broth while stirring to prevent lumps.

Add the potatoes, Cajun seasoning, smoked paprika, black pepper, and salt. Bring to a boil, then reduce the heat and simmer for 18 to 20 minutes, or until the potatoes are tender.

Use a potato masher to lightly mash some of the potatoes in the pot for a thicker soup, leaving plenty of chunks for texture.

Stir in the heavy cream and milk, then simmer gently for 3 to 5 minutes without boiling.

Remove the pot from the heat and stir in the cheddar cheese until melted and smooth.

Taste and adjust seasoning if needed. Serve hot, topped with green onions and parsley.