Start with the roasted garlic: Preheat the oven to 350 degrees F.

Peel the outermost layers of skin off the heads of garlic. Cut off the top 1/3 of the heads to open the cloves. Heat olive oil in a large oven-safe saute pan over medium heat. Add garlic, cut sides up, and saute for about 10 minutes. Season with salt and pepper. Transfer pan to the oven and roast for 15 minutes. Remove from heat and let cool. Pop garlic cloves from their skins and place cloves in a blender, along with any olive oil left in the pan. Puree until smooth; you should have a paste-like consistency.

For the potatoes: Cube the potatoes. Then put the potatoes in a large saucepan with salted cold water and place in the refrigerator overnight. The next day, add some more salt and then bring to a boil over high heat. Cook until tender, about 10 minutes. Drain well, place potato cubes in a food mill, and grind to remove skins. Alternatively, smash the potatoes with a large fork or potato masher.

Heat 1 tablespoon of the butter in a large saucepan over medium heat until the butter stops foaming and turns a light brown. Add the garlic paste and cook quickly. Add the cream, season, to taste, with salt and pepper, and bring to a boil. Reduce the heat to low and fold in potatoes with a wooded spoon or large whisk. Add the remaining butter by tablespoons, stirring after each addition. Stir in the extra-virgin olive oil. Season with salt and pepper, to taste, and serve.