In a large pot over medium heat, melt butter. Sauté onions until translucent. Increase heat to high, add potatoes and chicken stock (or vegetable stock), bring to a boil and cook until potatoes are very tender. Drain off excess liquid.
Carefully transfer potato mixture in batches to a blender and purée until very smooth. Return puréed soup to pot.
Whisk in milk and 1/2 cup cream. Season to taste with salt and pepper. Over medium heat, cook until small bubbles appear on surface of soup. Remove from heat, cover and refrigerate 3 to 4 hours.
Just before serving, whisk in remaining cream, dill and parsley. Ladle soup into bowls, garnish with green onions and serve.