A server cluster can be modeled as a set of parallel queues, and the dispatcher decides to which queue the arriving jobs are routed. We consider an energy-aware dispatching system in a Markovian setting, where each server, upon becoming empty, enters a sleep mode to save energy, and to activate the server after sleep incurs an additional setup delay cost. We seek to optimize the performance-energy trade-off by applying the so-called Whittle index approach. As our main result, we derive sufficient conditions for the system parameters under which the problem is provably indexable, and also determine the corresponding Whittle index values explicitly. Our numerical experiments demonstrate that the resulting energy-aware Whittle index policy is able to perform very close to the numerically solved optimal policy and outperforms all considered reference policies.