Put the eggs in a bowl and use a fork to whisk together, set aside.
Heat the oil with half the butter in a large skillet over a medium-high heat until frothy. Reduce the heat to medium, add the leeks, and stir around for about 5 minutes until wilted. Use a slotted spoon to remove the leeks from the skillet and set aside.
Add the remaining butter to the oil in the skillet. Season the eggs to taste with salt and pepper, then pour them into the skillet, reduce the heat to medium low and cook stirring for 1.5 minutes. or until the eggs begin to set.
Return the leeks to the skillet and stir into the eggs until the eggs have reached desired consistency.
Spoon the egg and leeks over the hot muffins and serve.