1. Place potatoes in a saucepan and cover with water. Bring to a boil and cook, partially covered, until tender (15-20 minutes).
2. While potatoes are cooking, combine leeks and milk in another saucepan. Bring to a simmer over low heat. Cook, partially covered, stirring occasionally, until the leeks are tender (15 minutes).
3. Drain the potatoes and return them to the pan. Mash until smooth. Stir in oil, leeks, and milk. Season with salt and pepper.