Add half of the butter to a non-stick frying pan (mine is 6 inches in diameter /15 cm).
Chop the mushrooms and add them to the pan with a pinch of salt and pepper.
Cook the mushrooms, stirring occasionally, until all the moisture released is absorbed and the mushrooms are golden.
Remove the mushrooms from the pan and set them aside.
In the same pan, add half of the remaining butter.
Crack the eggs in one bowl, and use a fork to beat them well.
Pour in half of the eggs, and gently swirl the pan around so the eggs can spread evenly.
Cook the eggs until they are slightly set, then sprinkle over half of the mushrooms and grated cheese.
Once the omelette is mostly set, carefully flip it over with a spatula and cook it for a further 30-40 seconds.
Remove it from the heat, and transfer it to a plate.
Make another omelette with the remaining ingredients.
Top the omelettes with extra cheese and chopped dill if you like.