Wrap the grated potatoes in a large clean towel and squeeze out the excess starch. Get them as dry as possible. Unwrap and put the potatoes in a large mixing bowl. Add 2 teaspoons of the salt, 1 teaspoon of the white pepper, and the olive oil and toss to mix well. Form the mixture into 12 small cakes about 3 inches in diameter and 1 inch thick.

Heat the vegetable oil in a 10- or 12-inch nonstick skillet over medium-high heat until it's hot but not smoking. Fry the cakes, several at a time, until they are golden brown and cooked through, about 4 minutes on each side. Drain on paper towels and keep warm.

Pour off the oil and wipe the skillet clean. Set aside.

In a large mixing bowl, whisk the eggs until frothy, about 1 minute. Add the remaining teaspoon salt, the remaining 1/4 teaspoon white pepper, the truffle oil if using, half-and-half, green onions, and parsley. Whisk to blend.

In the skillet, heat the butter over medium heat for 1 minute. Add the egg mixture. Using a wooden spoon, stir the eggs and cook until the mixture sets slightly, but is still soft, 7 to 8 minutes. Cook 2 to 3 minutes longer if firmer eggs are desired. Add the cheese, stir gently for about 30 seconds, and remove from heat. Continue stirring the mixture until the cheese melts completely.

To serve, top each potato cake with an equal portion off the eggs. Garnish with shaved truffles if desired. Serve warm.