Heat the oven to 350 degrees Fahrenheit.
Put unpeeled garlic on a piece of foil. Drizzle with about 1 teaspoon of the oil, season with salt and pepper. Fold the foil in to form a neat little package. Roast for about 30 minutes, until the garlic is soft but not caramelized.
Boil the potatoes until they are very tender, and drain. Let cool slightly.
Puree the one garlic clove into a smooth paste, using a mortar and a pestle. Add a pinch of salt and place in a bowl.
Cut off the heads of the roasted garlic cloves with scissors and squeeze the pulp into the bowl. Combine with the raw garlic. Drizzle in the wonderful oil from the foil package also.
When the potatoes are cool enough to handle, but still warm, peel them and add one at a time to the garlic. You can fluff them in with a fork and mix until smooth.
Drizzle in the remaining olive oil a bit a time, making sure to have a smooth puree before adding more oil. Season with the lemon juice, salt and black pepper.
If you want a looser sauce, add some warm water or chicken stock.
Serve right away, or keep at room temperature for few hours. If you need to keep it longer than that, refrigerate but return to room temperature before serving.