In a large pot add about 2-3 tablespoons of oil until hot and add the onions, carrots, and celery, and sauté until everything is coated with oil and they start to cook down (3 minutes). Add the split peas, 7 cups of water and bring to a rapid simmer. Simmer, uncovered until the peas have broken down, about 1 hour.
Once the peas are soft, use a hand infuser (or a blender) and blend soup until it forms a smooth consistency. Return soup to pot and add enough salt to season (start with 2 teaspoons), add milk and enough chicken stock to ensure soup is creamy but not too thick. Finally add parsley, and serve with home-made croutons.
For a heartier version of the soup, add 1 cup cooked barley to the soup or any grain of your choice.