Pinch off bits of sausage into a large saucepan set over medium heat. Cook sausage pieces until browned, about 6 minutes. Remove to a bowl and set aside, leaving fat in the saucepan. There should be about a tablespoon of fat; if less add vegetable oil to make up the difference. (Remove any fat in excess of 1 tablespoon.)
Add onion and salt to pan and cook until onion is softened, 6-8 minutes, stirring frequently. Add garlic and cook for an additional 30 seconds. Pour in vermouth or white wine and scrape any browned bits from bottom of pan.
Add potatoes, broth, water, allspice and bay leaf to pan. (You may have to add additional water if potatoes are not covered.) Bring to a simmer over medium-high heat. Reduce heat to medium and simmer until potatoes are tender, about 10 minutes.
Turn heat to low. Remove bay leaf and add cream and kale. Cook, without boiling, for 2-3 minutes or until soup is heated through and kale has started to soften. Remove from heat and add salt and pepper to taste. Serve immediately.