Add the olive oil to a skillet and heat it over medium-high heat. Whisk the eggs until they are frothy. Stir half of the butter into the eggs. Add the remaining butter to the skillet and swirl it with the oil until it melts and becomes cloudy and bubbly.
Pour the eggs into the hot skillet and cook, moving a fork quickly through the eggs in small circles and zigzags until the eggs are approximately 80% cooked through. Smooth down the top surface of the eggs with the back of a large spoon or a small offset spatula.
Season the eggs with salt and pepper to taste. Sprinkle the shredded cheese and chopped herbs on the eggs and cover with a lid. Turn off the heat and allow the omelet to continue cooking for 2 to 3 minutes, depending on how firm you want your eggs.
Tilt the skillet to the side a bit and, using a rubber spatula, carefully ease the omelet out of the pan and onto a warmed serving plate. Gently roll the omelet into the traditional tube shape.
This basic French omelette recipe makes 1 serving.