Since I moved to Innsbruck, this was on my mind. I finally went for it.
I started at ca. 5:10 at the Innbrücke, Innsbruck. In complete darkness, I hiked/ran to Achselboden where I met the sun (I filled up on water at the spring) and from there to Achselkopf. After that, I climbed the Brandjochspitze Südgrat and started to collect all peaks of the Nordkette heading east, all the way to Hochmadhkopf.
I managed to do the Via Ferrata really early, so there was no traffic. The only chance to restock my supplies was the Hafelekar topstation, where I filled up on water at the toilette and bought a Coke. After that, there is no spring on the route. I carried 4 soft flasks with around 2 liters, which was barely enough. After long hours of scrambling and running/hiking on the Nordkette, I made the steep descent to Absam, Tirol. After reaching the endpoint, I took the bus back to Innsbruck. This was one of my most mesmerizing days in the mountains I ever had.
It was a complete solo adventure.