Melt the butter in a heavy-bottomed pot, such as a Dutch oven. Brown the sausage on all sides (about 3 minutes). (If your pot doesn't fit all the meat in a single layer, do this step in batches.) Using a slotted spoon, transfer the sausage to a plate lined with a paper towel and set aside.
Spoon out and discard all but 1 teaspoon of the fat that's in the pot. Add onion and celery and saute until starting to soften (about 3 minutes). Add garlic and smoked paprika and stir around.
Add chicken broth and water, cover and bring to a simmer. Add the broccoli and the browned meat. Cover and simmer until the broccoli is tender and the meat is fully cooked (about 5 minutes).
Stir in oregano, cream and Parmesan. Stir in liquid smoke to taste.
