1. Bring potatoes and cold water to cover by 2 inches to boil in a large saucepan. Add 1 teaspoon salt and cook until tender, 12 to 15 minutes. Drain and press potatoes into a large bowl with a potato ricer or food mill. Beat in yolks, cheese, green onions, pine nuts, parsley, mint, remaining 3/4 teaspoon salt, and pepper.
2. Heat oven to 200 degree F. Line 2 jelly-roll pans with foil. Shape potato mixture by rounded tablespoons into 2-inch-long ovals. Place ovals on prepared pans.
3. Heat oil to 375 degrees F in a deep 10-inch frying pan over medium heat. Beat egg whites in bowl until frothy. Spread flour on a plate and bread crumbs on another plate. Dip ovals in flour, then egg whites, then coat in crumbs. Fry ovals, 5 or 6 at a time, until golden, 3 minutes. (Keep oil temperature between 325 degree F and 350 degree F.) Transfer croquettes with slotted spoon to a paper-towel-lined cookie sheet; keep warm in oven. Transfer to serving dish. Serve immediately. Makes 30 croquettes.