Heat the olive oil and half the butter in a non-stick frying pan.
Add the flat and chestnut mushrooms and fry for 3-5 minutes, stirring now and then, until lightly browned.
Remove the mushrooms from the pan and set aside.
Break the eggs into a bowl and beat together. Roughly shred half the arugula leaves and mix into the beaten eggs. Season well with salt and freshly ground pepper.
In a separate bowl, lightly beat the cream cheese. Mix in the sun-blushed tomatoes and half the cooked mushrooms. Season the cream cheese mixture with salt and freshly ground pepper and set to one side.
Heat the remaining butter until foaming in a large non-stick frying pan. Pour the beaten egg mixture into the pan, tilting the pan to spread it evenly, and fry until the omelette is set underneath but slightly runny on top.
Remove the pan from direct heat and spread the cream cheese mixture over the centre of the omelette. Using a palette knife, flip over each side of the omelette over the cream cheese filling and transfer onto a warm plate.
Meanwhile, divide the remaining arugula and the herb salad leaves among two plates.
Cut the filled omelette on a slant into 4 even-sized portions and place 2 omelette portions each on the arugula and salad. Top the omelette portions with the remaining fried mushrooms, season with freshly ground pepper and serve at once.