Beat the eggs with the filling of your choice.
Heat a crepe pan or heavy based frying pan/skillet with the butter and oil. Once the pan is hot, pour in the egg mixture swirling quickly to get an even and thin coating in the pan. Let the omelette cook for a couple of minutes over medium-high heat.
Lift the edge of the omelette with a spatula to check it is set and golden underneath; the top of the omelette should be just-about-set but still a little gooey.
Slip the omelette out of the pan onto a plate and flip 1/2 of the omelette over the other, or fold in 3 like a business letter. Carry on!