Crack the eggs into a bowl, add the milk, and whisk to blend with a fork.
Heat a nonstick pan over medium heat. (I use a 6-inch skillet.) Once the pan is hot, add enough olive oil or butter to lightly coat the surface of the pan.
Pour the eggs into the skillet and allow to cook without stirring until the bottom begins to set. With a heat-resistant rubber spatula, gently push one edge into the center of the pan while tilting the pan to allow the still-liquid egg to flow underneath. Repeat with the other edges until there is no liquid remaining and the top of the omelet is just slightly wet. If the omelet is cooking too fast or browning on the bottom, reduce the heat to medium-low.
Lay the cheese slices over one half of the omelet, breaking to create an even layer. Sprinkle the chives and basil overtop. Season with salt and pepper. (If you wish to add other filling ingredients, now is the time to do it.)
With your spatula, lift the half of the omelet without the toppings and fold it over, so that the edges line up. It helps to hold up the pan and tilt it at the same time to help get the half over in one piece. (If it breaks, don’t worry. It will still taste great!) Cook for another minute or so until the cheese is thoroughly melted and the omelet is cooked through.
Sprinkle with extra herbs, season to taste, cut in half, and serve immediately.