Preheat the oven to 350 degrees F.
Place the unpeeled garlic cloves in a small ovenproof container and cover with the olive oil. Cover with a lid or aluminum foil. Place in the oven and roast for 20 minutes, or until the cloves are tender when tested with the tip of a knife. Remove the garlic cloves and place on a plate to cool. Strain the olive oil and reserve for later use. Squeeze the roasted garlic out of the skins when cool enough to handle. Puree the garlic by chopping and smashing it with the blade and side of a knife, or with a mortar and pestle.
Meanwhile, peel the potatoes and cut them into 2-inch pieces. (There will be about 8 cups potatoes.) Place them in a large saucepan. Add enough water to cover the potates by 1-inch. Sprinkle with 1 teaspoon salt, place the pan over high heat, and bring to a boil. Reduce the heat, cover the saucepan with lid placed slightly ajar, and simmer the potatoes for 20 to 30 minutes, or until tender when pierced with the tip of a knife.
Combine the milk and goat cheese in small saucepan and heat to a simmer, stirring until there are no lumps. When the potates are cooked, drain them, return them to the pan, and place in the still-warm oven, uncovered to dry out for 10 minutes.
Remove the potatotes from the oven and pass them, along with the garlic, through a potato ricer or a food mill fitted with the fine disk. Alternatively, the potatoes can be whipped using an electric mixer. Add the reserved garlic oil and the goat cheese-milk mixture and whip the potatoes with a mixer or by hand, adding additional milk if too dry. Reheat, covered, over very low heat if necessary. Serve hot.