  Vortex $\gamma$ photons, which carry large intrinsic orbital angular momenta
(OAM), have significant applications in nuclear, atomic, hadron, particle and
astro-physics, but their production remains unclear. In this work, we
investigate the generation of such photons from nonlinear Compton scattering of
circularly polarized monochromatic lasers on vortex electrons. We develop a
quantum radiation theory for ultrarelativistic vortex electrons in lasers by
using the harmonics expansion and spin eigenfunctions, which allows us to
explore the kinematical characteristics, angular momentum transfer mechanisms,
and formation conditions of vortex $\gamma$ photons. The multiphoton absorption
of electrons enables the vortex $\gamma$ photons, with fixed polarizations and
energies, to exist in mixed states comprised of multiple harmonics. Each
harmonic represents a vortex eigenmode and has transverse momentum broadening
due to transverse momenta of the vortex electrons. The large topological
charges associated with vortex electrons offer the possibility for $\gamma$
photons to carry adjustable OAM quantum numbers from tens to thousands of
units, even at moderate laser intensities. $\gamma$ photons with large OAM and
transverse coherence length can assist in influencing quantum selection rules
and extracting phase of the scattering amplitude in scattering processes.
