Whisk the Eggs - In a medium bowl, whisk together the eggs, milk (if using), and salt. Whisk using a side-to-side motion until uniform in appearance, do not over mix.
Cook the Omelette -Method 1) American-StyleIn an 8-inch nonstick skillet, heat the butter over medium-low heat until it just begins to bubble and foam, but does not brown. Add the eggs to the pan, use a spatula to immediately pull the cooked curds from the pan's edges toward the center, allowing the raw egg to run underneath, tilting the pan to help move it to the bottom. Start from the top of the pan and create curds around the entire omelet, about 1 to 1 ½ minutes. Once the eggs are mostly cooked, similar to soft scrambled eggs with a glossy and slightly wet surface, turn the heat down to low. Sprinkle the cheese on one half of the omelet. Using a spatula, fold the other half on top. The surface should be golden brown. Method 2) French-StyleIn an 8-inch nonstick skillet, heat the butter over medium-low heat untilit just begins to bubble and foam, but does not brown.Add the eggs to the pan, then wait for 10-seconds until the edges begin to set. Use a spatula to pull the cooked curds from the pan's edges toward the center, allowing the raw egg to run underneath. Tilt the pan as needed. Start from the top of the pan, and work to create curds around the entire omelette, about 2 to 2 ½ minutes. Once the eggs are mostly cooked, with a glossy surface, turn the heat off. Using a spatula, fold one side to the center, then the other side, creating a rolled cigar shape. The surface should not be browned.
To Serve - Slide the American-style omelet onto a plate. For the French-style omelette, gently invert it onto a plate seam-side down to keep its smooth, delicate shape. If desired, sprinkle cheese on top while warm so it slightly melts. Garnish with freshly cracked black pepper and chopped chives.
Notes
Adding Liquid: This makes the egg softer and more tender but lessens the egg flavor. Dairy also adds extra richness.
Saltier Eggs: Use ¼ teaspoon of kosher salt instead of 1/8 teaspoon.
Less Browned Surface: For the American-style omelet, use medium instead of medium-high heat.
For Cooking 3 to 4 Eggs: Use a 10-inch skillet and look for the appearance change. The cook time will be seconds.
Adjusting Batch Size: For every large egg, use ½ teaspoon butter or oil and 1/16 to ⅛ teaspoons of salt. See pan size recommendations in the full article.