Crack eggs into a medium bowl and season with salt. Whisk until smooth.

In a medium nonstick skillet over medium-high heat, heat 1 teaspoon oil. Add half the egg mixture and swirl pan to evenly coat bottom of skillet with egg. Cook, undisturbed, until edge is set, about 1 minute.

Run a spatula along edge of egg to release, then use your hands to gently flip. Cook until egg is just set on the bottom, about 20 seconds.

Slide omelet onto a cutting board and let cool 1 minute. Meanwhile, repeat process with remaining oil and egg. Roll omelets like a cigar and slice into ¼"-inch-thick noodles.

Return skillet to medium-high heat and melt butter. Add garlic and cook until fragrant, 1 minute. Add noodles, Parmesan, and parsley and toss.

Season with pepper and sprinkle with Parmesan before serving.