Motivated by observations of a strong near-inertial wave signal at the base of the semi-permanent anticyclonic Cyprus eddy during the 2010 BOUM experiment, a numerical study is performed to investigate the role of near-inertial/eddy interactions in energy transfer out of the mixed layer. A hybrid temporal-spatial decomposition is used to split all variables into three independent components: slow (eddy) and fast (inertial oscillations + waves), which proves useful in understanding the flow dynamics. Through a detailed energy budget analysis, we find that the anticyclonic eddy acts as a catalyst in transferring wind-driven inertial energy to propagating waves. While the eddy sets the spatial scales of the waves, it does not participate in any energy exchange. Near-inertial propagation through the eddy core results in the formation of multiple critical levels with the largest accumulation of wave energy at the base of the eddy. A complementary ray-tracing analysis reveals critical level formation when the surface-confined inertial rays originate within the negative vorticity region. In contrast, rays originating outside of this region focus at the base of the eddy and can propagate at depth.