Heat 2 teaspoons of oil in a non-stick pan. Add the garlic and zucchini and sauté over medium–high heat until lightly golden, shuffling them with a wooden spoon so that they cook evenly. Season with salt and mix in the basil. Remove from the heat and remove the garlic.
Whisk one egg in a bowl. Heat 1 teaspoon of oil in a 14 cm(6 inch) non-stick frying pan over medium–low heat. Add the egg, swirling it evenly around the pan. Scatter about a quarter of the zucchini over the omelette and about 1 tablespoon of parmesan. Cook until the omelette is set and the underside is lightly golden. Slip out onto a warm plate while you make the rest, adding more oil to the pan as needed. Serve with a few drops of lemon juice and a grinding of pepper if you like.