Enhancement cavities (ECs) seeded with femtosecond pulses have developed into the most powerful technique for high-order harmonic generation (HHG) at repetition rates in the tens of MHz. Here, we demonstrate the feasibility of controlling the phase front of the excited transverse eigenmode of a ring EC by using mirrors with stepped surface profiles, while maintaining the high finesse required to reach the peak intensities necessary for HHG. The two lobes of a TEM01 mode of a 3.93m long EC, seeded with a single-frequency laser, are delayed by 15.6 fs with respect to each other before a tight focus, and the delay is reversed after the focus. The tailored transverse mode exhibits an on-axis intensity maximum in the focus. Furthermore, the geometry is designed to generate a rotating wavefront in the focus when few-cycle pulses circulate in the EC. This paves the way to gating isolated attosecond pulses (IAPs) in a transverse manner (similarly to the attosecond lighthouse), heralding IAPs at repetition rates well into the multi-10MHz range. In addition, these results promise high-efficiency harmonic output coupling from ECs in general, with an unparalleled power scalability. These prospects are expected to tremendously benefit photoelectron spectroscopy and extreme-ultraviolet frequency comb spectroscopy.