The optimal enhancement of broadband optical pulses in a passive resonator requires a seeding pulse train with a specific carrier-envelope-offset frequency. Here, we control the phase of the cavity mirrors to tune the offset frequency for which a given comb is optimally enhanced. This enables the enhancement of a zero-offset-frequency train of sub-30-fs pulses to multi-kW average powers. The combination of pulse duration, power, and zero phase slip constitutes a crucial step toward the generation of attosecond pulses at multi-10-MHz repetition rates. In addition, this control affords the enhancement of pulses generated by difference-frequency mixing, e.g., for mid-infrared spectroscopy.