Even after decades of software engineering research, complex computer systems still fail, primarily due to nondeterministic bugs that are typically resolved by rebooting. Conceding that Heisenbugs will remain a fact of life, we propose a systematic investigation of restarts as "high availability medicine." In this paper we show how recursive restartability (RR) - the ability of a system to gracefully tolerate restarts at multiple levels improves fault tolerance, reduces time-to-repair and enables system designers to build flexible, highly available software infrastructures. Using several examples of widely deployed software systems, we identify properties that are required of RR systems and outline an agenda for turning the recursive restartability philosophy into a practical software structuring tool. Finally, we describe infrastructural support for RR systems, along with initial ideas on how to analyze and benchmark such systems