In reply to Russell Lovett:
You could follow the GR10 on the French side through the Parc National des Pyrenees which is roughly from the Cirque de Lescun (west) to Cirque de Troumouse (east) or the Pyrenean Haute Route (HRP) which sticks closer to the border and crosses it in places through the same stretch.
You'd have to work out for yourself exactly how long that would take you but at a guess it must be about a week and would incorporate many of the scenic highlights. There are regular huts and most have a camping area or you could wildcamp.