Mix together the goats' cheese with the thyme and the olives in a salad bowl. Season with salt and pepper and bind with the egg. Set aside.
Separate the filo pastry sheets. Brush each one with the oil, then cut into 4 strips.
Take the pastry strips one at a time and place ½ teaspoon of filling about 2.5 cm (1 inch) from the nearest end. Fold the pastry over and over to wrap the filling in a triangle, then seal the end with a little egg white. Repeat with all the strips.
Heat the oil until a cube of bread browns in 30 seconds and deep-fry the pastries, in batches, until golden on both sides. Drain on kitchen paper.
Serve hot, with lettuce leaves if you wish.