Place the unpeeled garlic cloves in an 8-inch skillet over the lowest possible heat and cook until the skins are spotty dark brown and the cloves are slightly softened, about 22 minutes.

Remove the skillet from the heat, cover it tightly, and let the garlic stand until the cloves are completely soft, 10 to 15 minutes.

Peel the softened cloves—the skins should slip right off easily—and mince them into a smooth paste.

Place the potatoes in a large pot and cover with 1 inch of cold water. Add the salt and bring to a boil over high heat, then reduce to a simmer and cook until a paring knife slides easily through the chunks, about 15 minutes.

Drain the potatoes thoroughly and return them to the pot over low heat for 1 minute to evaporate any remaining surface moisture. Use a potato ricer, food mill, or hand masher to process the potatoes until no large lumps remain.

Fold the garlic paste, butter, and warm milk into the potatoes. Stir gently until the butter is melted and the texture is creamy and uniform.

Serve warm, with an extra pat of butter if desired.