  In this paper we investigate the regularity and solvability of solutions to
Dirichlet problem for fully non-linear elliptic equations with gradient terms
on Hermitian manifolds, which include among others the Monge-Amp\`ere equation
for $(n-1)$-plurisubharmonic functions. Some significantly new features of
regularity assumptions on the boundary and boundary data are obtained, which
reveal how the shape of the boundary influences such regularity assumptions.
Such new features follow from quantitative boundary estimates which
specifically enable us to apply a blow-up argument to derive the gradient
estimate. Interestingly, the subsolutions are constructed when the background
space is moreover a product of a closed Hermitian manifold with a compact
Riemann surface with boundary.
