Snap off the woody ends of the asparagus and cut the spears into 5 cm (2 inch) lengths. Slice the potatoes very thinly.
Heat 50 ml (2 fl oz) of the oil in a sturdy frying pan about 25 cm (10 inch) across. Add the asparagus and fry gently for 5 minutes until slightly softened. Lift out with a slotted spoon on to a plate. Add the remaining oil to the pan and scatter in the potatoes and onion. Cook very gently, turning frequently in the oil, for about 15 minutes until the potatoes are tender.
Beat the eggs with a little salt and pepper in a bowl and stir in the basil leaves. Add the asparagus to the pan and combine the vegetables so that they are fairly evenly distributed. Pour the egg mixture over the vegetables and reduce the heat to its lowest setting. Cover with a lid or foil and cook for about 10 minutes until almost set but still a little wobbly in the centre.
Loosen the edge of the tortilla, cover the pan with a plate and invert the tortilla on to it. Slide back into the pan and return to the heat for 2–3 minutes until the base has firmed up. Slide on to a clean plate and serve warm or cold, cut into wedges.