We present the first computation of the NLO photon+dijet impact factor in e+A DIS at small x. When combined with the extant results for the JIMWLK small x evolution to NLLx accuracy, this result provides us with a prediction of the photon+dijet cross-section in e+A DIS to O( (\alpha_S)^3 ln(1/x) ) accuracy. The comparison of this result with photon+dijet measurements at a future EIC therefore provides a precision test of the systematics of gluon saturation. In the soft photon limit, one obtains a compact representation of the state-of-the art results for fully inclusive DIS. The novel techniques developed in this computation can also be applied to promote existing LO computations of photon+dijet production in p+A collisions to NLO+NLLx accuracy.