Add 3 tablespoons of the butter to a medium skillet, over medium heat, until melted. Add the leeks and cook until the leeks are broken down and softened, about 10 minutes. Remove from the heat and set aside.
Melt the remaining butter in large saucepan, over medium heat. When the butter foams, add the potatoes and onions and stir to coat with the butter. Add salt and pepper, to taste, and cover with a lid. Reduce the heat and let cook for 8 minutes without browning. In a separate saucepan, bring the stock to the boil over medium heat. Add the stock to the potato mixture and simmer for 5 minutes. Let the soup cool for about 5 minutes, then add the milk. Carefully transfer the soup to a food processor and pulse until smooth. Pour the soup into serving bowls, then spoon the reserved leeks on top. Sprinkle with chives and parsley and serve.
This recipe was provided by a chef, restaurant or culinary professional. It has not been tested for home use.