Waltuck likes to use a cast-iron omelet pan that's well seasoned, but you can also use a sauté pan. He strongly recommends using clarified butter, which keeps for weeks in the fridge (see below to learn how to make it).
To make an omelet for one person, break three eggs into a bowl, add a tablespoon of milk, and salt and pepper to taste; whisk until combined. Heat the omelet pan until very hot. Then add a tablespoon of clarified butter, swirling it around in the pan to coat the bottom and sides evenly.
Add the egg mixture and immediately begin stirring with a fork so that you're kind of scrambling the eggs from the bottom of the pan. Turn the heat down after the bottom of the eggs begins to solidify, and add desired fillings, such as chopped vegetables, cheese, and meat. Let the omelet set for a few seconds, then turn off the heat and roll the omelet. It should be smooth, egg colored, and a little runny in the center.
To make clarified butter: In a separate pan, heat butter until it melts, then turn off the heat. Skim off what's on top with a large spoon, save the clear butter, and discard the milk solids. The benefit of clarified butter is that it's purer fat and burns at a higher temperature, which is ideal for an omelet.