Was my initial thought but would be tough (or perhaps just not much fun) to fit into 10 days with travel either side. Also quite exposed in places.
I’ve done it in two trips starting at the midpoint, Vizzavona, both times. Heading North the first time and South the second time.
The Northern section was by far the harder and more exposed. The Southern end is far less ‘rugged’ and exposed although still fairly tough. You could do one half in a ten day trip perhaps. There’s a mountain railway to get you up to Vizzavona from Ajaccio.