Whisk together eggs, pecorino, ½ teaspoon black pepper, and ⅛ teaspoon salt in a medium bowl until well combined and slightly frothy; set aside.
Melt butter in a 10-inch nonstick skillet over medium; swirl skillet to evenly coat bottom.
Holding potato over skillet, use a peeler to peel the potato into long, wide ribbons directly into skillet until you have enough ribbons to evenly cover bottom of skillet (you may not use the entire potato). Gently spread potato into an even layer.
Sprinkle evenly with ¼ teaspoon pepper and ⅛ teaspoon salt. Cover and cook, undisturbed, until potato ribbons are tender and starting to brown on bottom, about 3 minutes.
Pour egg mixture evenly over potato ribbons. Cover, reduce heat to low, and cook, undisturbed, until eggs are mostly set, 4 to 5 minutes.
Uncover and sprinkle fontina evenly over top. Using a spatula, loosen omelet around edges and fold in half.
Cook until cheese is melted and omelet is set but still tender in center, 30 seconds to 1 minute.
Slide omelet onto a plate. Garnish with additional pecorino Romano and black pepper. Serve immediately.
