We introduce a novel hybrid microwave amplifier that utilizes the axial bunching mechanism of klystrons in conjunction with the energy extraction mechanism of cyclotron resonance masers. A simple analytic model is used to show the viability of the device configuration and to explore limitations of the scheme. An example of an 11 GHz, 1 MW, fourth harmonic amplifier is presented. Adopting a single-particle approach, numerical simulations are used to demonstrate bunching in physical space and to estimate system efficiency. It is shown that this configuration enables the design of efficient, high harmonic devices over a wide range of parameters.