Two effects, jet broadening and gluon bremsstrahlung induced by the propagation of a highly energetic quark in dense QCD matter, are reconsidered from effective theory point of view. We modify the standard Soft Collinear Effective Theory (SCET) Lagrangian to include Glauber modes, which are needed to implement the interactions between the medium and the collinear fields. We derive the Feynman rules for this Lagrangian and show that it is invariant under soft and collinear gauge transformations. We find that the newly constructed theory SCET G recovers exactly the general result for the transverse momentum broadening of jets. In the limit where the radiated gluons are significantly less energetic than the parent quark, we obtain a jet energy-loss kernel identical to the one discussed in the reaction operator approach to parton propagation in matter. In the framework of SCET G we present results for the fully-differential bremsstrahlung spectrum for both the incoherent and the Landau-Pomeranchunk-Migdal suppressed regimes beyond the soft-gluon approximation. Gauge invariance of the physics results is demonstrated explicitly by performing the calculations in both the light-cone and covariant R ξ gauges. We also show how the process-dependent medium-induced radiative corrections factorize from the jet production cross section on the example of the quark jets considered here.