Cacio E Pepe Egg Noodles

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 twirl the pan to evenly coat the bottom of skillet with egg. Cook, undisturbed, until edges are set, about 1 minute.  Run a spatula along edges of egg to release, then use your hands to gently flip the omelette. Cook until egg is just set on the bottom, about 20 seconds. Slide omelette onto a cutting board and let cool slightly for 1 minute. Meanwhile, repeat process with remaining oil and eggs. Roll omelettes up like a cigar and slice into ¼-inch thick "noodles." Return skillet to medium-high and melt butter. While whisking, slowly pour in ¼ cup water until combined. Add Parmesan and pepper, and stir until Parmesan melts into sauce. Add noodles and toss to coat.  Season with more pepper and sprinkle with more Parmesan before serving.