Cut the stalks into 2 cm (¾ in) slices at an angle. Blanch the stalks in boiling salted water for 1 minute. Blanch the tips for seconds only. Drain well.
Bring the cream to the boil in a large saucepan with the whole garlic cloves and simmer until the cloves are soft. Remove from the heat, and discard the garlic.
Heat the oil and butter in a separate saucepan and fry the remaini