Spin Hall oscillators (SHOs) based on bilayers of a ferromagnet (FM) and a non-magnetic heavy metal (HM) are electrically tunable nanoscale microwave signal generators. Achieving high output power in SHOs requires driving large-amplitude magnetization dynamics by a direct spin Hall current. The maximum possible amplitude of such oscillations with the precession cone angle nearing 90 • is predicted for FM layers with easy-plane magnetic anisotropy and spin Hall current polarization perpendicular to the easy plane. While many FMs exhibit natural easy-plane anisotropy in the FM film plane, the spin Hall current in a HM|FM bilayer is polarized in this plane and thus cannot drive large-amplitude magneto-dynamics. Here we present a new type of SHO engineered to have the easy-plane anisotropy oriented normal to the film plane, enabling large-amplitude easy-plane dynamics driven by spin Hall current. Our experiments and micromagnetic simulations demonstrate that the desired easy-plane anisotropy can be achieved by tuning the magnetic shape anisotropy and perpendicular magnetic anisotropy in a nanowire SHO, leading to a significant enhancement of the generated microwave power. The easy-plane SHO experimentally demonstrated here is an ideal candidate for realization of a spintronic spiking neuron. Our results provide a new approach to design of high-power SHOs for wireless communications, neuromorphic computing, and microwave assisted magnetic recording.