Crack the eggs into a small bowl and whisk vigorously for about 20 seconds until the mixture is slightly frothy and fully combined.
Add a pinch of flaky sea salt and freshly cracked black pepper to the eggs, then gently whisk again to incorporate.
Grate the sharp cheddar cheese using a fine grater, aiming for about 1/4 cup of evenly shredded cheese, and set aside nearby.
Heat your non-stick skillet over medium-low heat and add 1 teaspoon of unsalted butter. Wait until it melts and begins to bubble gently, giving off a toasty aroma and turning slightly golden.
Pour the whisked eggs into the skillet, tilting the pan to spread them out evenly. Let cook undisturbed for about 10-15 seconds until the edges just start to set and turn a light golden.
Gently push the cooked edges toward the center with a spatula, tilting the pan slightly to allow the uncooked eggs to flow to the edges. Continue this process for about 30 seconds until most of the surface is just barely runny and the eggs are fluffy.
Sprinkle the grated cheddar cheese evenly over one half of the omelette. Wait about 20 seconds until the cheese begins to soften and melt into the eggs.
Carefully fold the omelette in half with your spatula, covering the cheese. Let it cook for another 15-20 seconds until the cheese is gooey and the surface turns a light golden brown.
Gently slide the omelette onto a plate and let it rest for a minute to settle and allow the cheese to set slightly. Garnish with chopped chives or parsley if desired, then serve immediately.
