Place garlic in a souffle dish or ramekin. Drizzle with oil. Bake in a 350 degree F oven about 20 minutes or until garlic is very soft; cool. Peel the garlic, discarding skins and reserving oil.
Peel and cut up potatoes. Cook, covered, in boiling salted water for 20 to 25 minutes or until tender; drain. Transfer potatoes to a large mixing bowl. Beat potatoes with an electric mixer on low speed. Add sour cream, the 1/4 cup cheese, baked garlic, reserved oil, and salt. Gradually beat in enough milk to make the mixture fluffy. Fold in basil. Spoon the potato mixture into a greased 2-quart casserole.
Cover and bake in a 325 degree F oven for 40 minutes. Stir; sprinkle with the 3 tablespoons cheese and drizzle with butter or margarine. Continue baking, uncovered, for 10 to 15 minutes more or until heated through. Serves 10 to 12.
Prepare potatoes and spoon into casserole but do not bake. Cover and chill up to 24 hours. Bake, covered, in a 325 degree F oven for 45 minutes. Stir; sprinkle with the 3 tablespoons cheese and drizzle with butter or margarine. Return to oven; bake 30 to 45 minutes or until heated through.