In a medium bowl, whisk together the eggs, pecorino cheese, ½ teaspoon of black pepper, and ⅛ teaspoon of salt until the mixture is well combined and slightly frothy. Set this aside.
In a 10-inch nonstick skillet, melt the butter over medium heat, swirling the skillet to coat the bottom evenly.
Using a peeler, peel the potato into long, wide ribbons directly into the skillet, ensuring you cover the bottom of the skillet with the ribbons (you may not need to use the entire potato).
Gently spread the potato ribbons into an even layer and sprinkle them with ¼ teaspoon of black pepper and ⅛ teaspoon of salt.
Cover the skillet and cook the potato ribbons undisturbed until they are tender and starting to brown on the bottom, which should take about 3 minutes.
Pour the egg mixture evenly over the potato ribbons.
Cover the skillet again, reduce the heat to low, and cook undisturbed until the eggs are mostly set, about 4 to 5 minutes.
Uncover the skillet and sprinkle the shredded fontina cheese evenly over the top.
Using a spatula, gently loosen the edges of the omelet and fold it in half.
Cook for an additional 30 seconds to 1 minute, until the cheese is melted and the omelet is set but still tender in the center.
Carefully slide the omelet onto a plate and garnish with additional pecorino Romano cheese and black pepper. Serve immediately.
