Heat the oil in saucepan over medium heat. Add the garlic, onion, and celery and cook until translucent. Pour the chicken broth over the vegetables and bring to a boil. Add 3 cups of the frozen potatoes, the minced dill, salt, and pepper to the broth. Bring to a boil and then reduce the heat to a simmer. Simmer until the vegetables are very tender, about 15 minutes.
Place the cream cheese in a blender. Ladle a cup of the hot soup over the cheese and blend until smooth, adding more soup as needed until all of the lumps are gone. Pour the cheese mixture back into the soup in the pot and stir until blended.
Add the remaining 1 cup of potatoes to the soup. Stir in the milk and bring to a simmer for another 15 minutes. Garnish with a sprig of dill.