In a large pot, melt the butter over medium heat. Add the onions, carrots, and celery, and cook gently for 5–7 minutes, until softened. Season with salt and freshly ground black pepper.
While the vegetables are cooking, slice the cooked gizzards into strips and set them aside.
Stir in the garlic, marjoram, and bay leaf. Cook for about 1 minute, until fragrant.
Add the potatoes, then pour in 6 cups of gizzard cooking broth (or bone broth). Bring to a gentle simmer and cook for 15–20 minutes, until the vegetables are tender.
Stir in the sliced cooked gizzards and simmer for about 10 minutes, until warmed through.
In a small bowl or cup, whisk together the sour cream and flour until smooth. Slowly whisk in 1–2 ladles of the hot soup to temper the mixture. Gradually stir it back into the soup, then simmer gently for a few minutes until slightly thickened and silky.
Remove the bay leaf, then taste and adjust salt and pepper as needed. Let the soup rest for 10 minutes before serving if possible. Ladle into bowls and finish with a generous sprinkle of fresh parsley.