In a large pot or Dutch oven, melt 1 tablespoon of unsalted butter over medium heat. Add the bulk mild Italian sausage and cook until browned, about 5-7 minutes. Make sure to break it into bite-sized pieces as it cooks.
Once the sausage is browned, add the finely diced onion, diced celery, and minced garlic. Sauté for another 3-4 minutes, until the onion is translucent and fragrant.
Sprinkle in 1 teaspoon of smoked hot paprika and stir well to combine. This will add a delightful warmth and depth of flavor to the soup.
Pour in the 3 cups of chicken broth and 1 cup of water. Bring the mixture to a gentle boil.
Once boiling, add the small broccoli florets and dried oregano. Reduce the heat to a simmer and let cook for about 8-10 minutes, or until the broccoli is tender but still vibrant green.
Stir in the 1 cup of heavy cream and 1 cup of grated Parmesan cheese until melted and creamy. At this point, add 1 to 2 splashes of liquid smoke for an extra layer of flavor. Taste and adjust seasoning as needed.
Ladle the soup into bowls and serve warm. You can top it with additional Parmesan cheese or fresh herbs for garnish if desired.