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In a large bowl, lightly beat together the 5 egg yolks and 2 whole eggs. In a separate bowl, use an electric whisk to beat the egg whites until stiff. Add whites to the yolks and fold together carefully, keeping it light and fluffy. Fold in the artichokes, pepper, half the cheese, the basil, salt and pepper.
Heat grill to high. Heat a medium non-stick frying pan over a medium heat. Add the butter and oil. When the butter has melted, add the omelette mixture and spread evenly. Cook until golden underneath, about 5 mins. Scatter over remaining cheese, then place the pan under the grill and cook for a further 2 mins. Slide the omelette onto a board or serving plate. Cut into wedges and serve.