Peel and cube 2 large baking potatoes. Cover with cold, salted water in a saucepan. Bring to a boil; lower heat and simmer for 15 minutes. Drain, return to pot and steam for 1 minute.
Add salt and pepper; mash potatoes well in pot using a masher, or force through a ricer and into a bowl. Stir in 1 egg yolk and let cool. Put remaining egg white and another whole egg in a bowl; beat with a fork.
Put 1/4 cup all-purpose flour in a small bowl. Mix 1/2 cup plain dry bread crumbs, salt and pepper in another small bowl.
Form 1 Tbsp. potato mixture into a ball; flatten into a 1/2-inch-thick patty. Repeat with remaining mixture. Turn patties, one at a time, in flour, then egg, then bread crumbs and place on a plate.
Pour 3 cups vegetable oil into a large saucepan over medium-high heat. Insert a deep-fry thermometer; heat to 350°F. Cook half of croquettes, turning once, until golden brown, 2 to 3 minutes. Remove with a slotted spoon; drain on paper towels. Repeat with remaining croquettes.