Heat the grill to its highest setting and place the tomatoes on a square of foil underneath, turning occasionally to prevent burning. When the tomatoes are slightly scorched, remove from the grill, squashing them slightly to release the juices.
Break the eggs into a bowl and mix with a fork. Add a small splash of water and mix. Add the chives and some black pepper, and beat some more. Set aside while you prepare the mushrooms.
In a non-stick frying pan, heat the butter over a medium heat until foaming. Add the mushrooms and cook for 5-8 mins until tender, stirring every few mins. Remove and set aside.
Briskly stir the egg mixture, then add to the hot pan (tilting it so that the mixture covers the entire base) and leave for 10 secs or so until it begins to set. With a fork, gently stir the egg here and there so that any unset mixture gets cooked.
While the egg mixture is still slightly loose, spoon the mushroom mix onto one side of the omelette, and top with the cream cheese and basil leaves. Flip the other side of the omelette over to cover, if you like. Leave to cook for 1 min more, then cut in half and slide each half onto a plate. Serve immediately with the tomatoes on the side.