  Vortex $\gamma$ photons with intrinsic orbital angular momenta (OAM) possess
a wealth of applications in various fields, e.g.-strong-laser physics, nuclear
physics, particle physics and astrophysics-yet their generation remains
unsettled. In this work, we investigate the generation of vortex $\gamma$
photons via nonlinear Compton scattering of ultrarelativistic electrons in a
circularly polarized laser pulse. We develop a quantum electrodynamics
scattering theory that explicitly addresses the multiphoton absorption and the
angular momentum transfer mechanism. In pulsed laser fields, we unveil the
vortex phase structure of the scattering matrix element, discuss how the vortex
phase could be transferred to the radiated photon, and derive the radiation
rate of the vortex $\gamma$ photon. We numerically examine the energy spectra
and beam characteristics of the radiation, while also investigating the
influence of finite laser pulses on the angular momentum and energy
distribution of the emitted vortex $\gamma$ photons.
