Melt the butter in a large heavy pot over medium heat.
Add the sliced kielbasa in a single layer and cook for 4 to 5 minutes, turning occasionally, until the cut sides are browned.
Transfer the browned kielbasa to a plate and leave the rendered drippings in the pot.
Add the chopped onion and carrots to the pot and cook for 5 to 7 minutes, stirring occasionally, until softened.
Stir in the potatoes, broth, marjoram, salt, and black pepper.
Bring the soup to a boil, then reduce the heat and maintain a gentle simmer for about 25 minutes, or until the potatoes are tender when pierced with a knife.
Place the sour cream in a bowl and whisk in a spoonful of hot broth to temper it.
Lower the heat and stir the tempered sour cream into the soup until evenly combined.
Return the kielbasa to the pot and simmer gently for 5 minutes without allowing the soup to boil.
Taste and adjust the salt and pepper, then let the soup rest for 5 minutes before serving.
Ladle the soup into bowls, garnish with fresh dill, and serve warm with rye bread.