Cook the potatoes in salted water until soft. Force through a ricer or a medium sieve. Lightly whisk the egg yolks and add immediately, with the butter, while the potatoes are still warm. Taste and add more salt and butter if necessary. Cool to just above room temperature. Shape the mashed potato into logs about 1 inch (2.5 cm) in diameter and 3 inches (7.5 cm) long. Roll them in the egg white, covering all sides, and then in the breadcrumbs.
Cover and refrigerate until half an hour before frying.
Preheat the oven to 375°F (190°C). Heat the oil in a deep saucepan to 350° to 375°F (175° to 190°C).
Using a slotted spoon or Chinese sieve, gently lower 4 to 5 croquettes into the oil. The oil will bubble up. Deep-fry until the croquettes are a golden brown. This will happen quite quickly. Remove from the oil and place on a baking sheet lined with paper towel. Put them in the oven to keep warm. Repeat with the remaining croquettes. The croquettes can be kept in a warm oven for 20 minutes. Any more than that and you run the risk of the potatoes drying out.
COOK'S TIP
The uncooked croquettes must be removed from the refrigerator half an hour before cooking. If you fail to do so, the hot oil may lift the breadcrumb coating away from the potatoes.