Thoroughly wash all vegetables and pat dry before prepping. Potatoes peels can be left on or removed according to personal preference.
Brown 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 jalapeno pepper to 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, until heated through and potatoes are tender, approximately 20 minutes. Stir occasionally while cooking.
Transfer 3-4 cups of soup mixture to a blender or food processor and blend until smooth. (See safety tip above first!). Pour pureed soup mixture back into 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!