Thoroughly wash all vegetables and pat dry before prepping. Potatoes peels can be left on or removed according to personal preference.
Brown the sausage in a large, high-sided skillet over medium-high heat. Transfer sausage to a plate and drain all but 1-2 tablespoons of fat from skillet.
Reduce heat to medium and add chopped celery, carrots, onion, garlic, and jalapeño pepper to the skillet. Sauté for several minutes, stirring continually, until vegetables are soft and fragrant.
Return sausage to skillet, along with chicken or vegetable stock. Add potatoes, sweet corn, oregano, and bay leaf to skillet. Season with salt and black pepper, to taste, and stir to combine.
Reduce heat to medium-low and simmer, uncovered for 20 minutes, until heated through and potatoes are tender. Stir occasionally while cooking.
Transfer 3-4 cups of the soup mixture to a blender or food processor and blend until smooth. (*See safety tip below first!). Pour the pureéed soup mixture back into the skillet and stir to combine.
To prevent curdling, add 2-3 tablespoons of hot soup mixture, one tablespoon at a time, to half and half before stirring into the skillet. Stir to combine and simmer another 5-10 minutes to heat through. Remove bay leaf and serve immediately – enjoy!