Peel the potatoes and cut innto slices about 1/4 inch thick. Drop them into boiling water and bring rapidly back to a boil again. Boil for 3 minutes then drain.
Make roux: melt the butter in a saucepan and blend in flour. Cook for 2 minutes without browning. Remove from heat and when the roux has stopped bubbling, add the hot milk beating with wire whip until smooth. Blend in the cream, basil, garlic, salt and pepper.
Return mixture to moderately high heat and stir as sauce thickens and comes to a boil. Simmer 2 minutes. Fold in potatoes. Sauce should just cover the potatoes. Add more milk or cream if needed. Bring to a simmer, correct seasonings, cover and simmer slowly for 10 - 15 minutes or until potatoes are tender.
Before serving, gently fold in the butter and garnish with additional basil.