Melt butter in a soup pot over medium heat. Add sausage and cook and stir until lightly browned, about 3 minutes. Use a slotted spoon to remove sausage to a bowl, leaving behind any butter in the pot.
Toss onion into the pot with 1 teaspoon salt. Cook and stir until onion just starts to turn translucent and picks up a brown color from the meat juices in the pan, 4 to 5 minutes. Add flour; cook and stir for 2 to 3 minutes. Add garlic and paprika; cook and stir until fragrant, about 1 minute, being careful not to burn the paprika.
Stir in 5 cups chicken broth. Increase heat to high and bring to a simmer, stirring occasionally so flour does not stick to the bottom of the pot. Stir in cabbage and cook until it just loses its stiffness, about 2 minutes. Stir in potatoes and bring back to a simmer. Add black pepper, cayenne pepper, and bay leaf, plus a splash of chicken broth if desired. Bring back to a simmer.
Stir soup and reduce heat to medium-low. Simmer, stirring occasionally, until potatoes are very soft and tender, about 30 minutes. Add sausage and vinegar and simmer for 10 minutes. Taste and adjust for salt.
Remove from heat and ladle into bowls. Garnish with sour cream, green onions, and cayenne pepper.
Olive oil can be used for butter, apple cider vinegar or white wine vinegar can be used for white vinegar, and any paprika can be used in place of Hungarian paprika.
If you can’t find Hungarian sausage, use Polish sausage or any other smoked sausage. Use chives, parsley, or dill to garnish the soup instead of green onions, if preferred.
Cut your potatoes into any size you want but pieces should be uniform and small enough to fit on a soup spoon.
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