Melt the butter in a large heavy-bottomed pot or Dutch oven over medium heat. Add the chopped onion and sauté for 4 to 5 minutes until translucent and fragrant.

Pour the fish or vegetable broth into the pot and bring to a gentle boil. Carefully add the diced potatoes, sliced carrots, bay leaves, and whole black peppercorns.

Reduce the heat to medium-low, partially cover the pot, and let the soup simmer for 15 to 20 minutes, or until the potatoes and carrots are easily pierced with a fork.

Gently lower the white fish chunks into the simmering broth. Cook uncovered for 5 to 7 minutes, or until the fish is completely opaque, flaky, and cooked through. Avoid stirring too vigorously so the delicate fish pieces do not break apart.

Remove the pot from the heat and carefully fish out and discard the bay leaves. Stir in the fresh lemon juice and chopped dill, then season generously with salt and black pepper to taste.

Ladle the hot soup into deep bowls, ensuring everyone gets a generous mix of the tender fish and root vegetables. Serve immediately with a slice of crusty dark rye bread and a little extra fresh dill on top for garnish.