We propose a fully differentiable end-to-end (E2E) generalized echo and interference cancellation (GEIC) backend that unrolls a generalized sidelobe canceller (GSC) decomposition in beamformer (BF)-to-acoustic echo canceller (AEC) processing order. The system comprises mask-driven steering tracking, a null-space projection blocking matrix, and dual-branch adaptation consistent with the convolutive transfer function (CTF) model using normalized least mean squares (NLMS). A single gated recurrent unit (GRU) controller jointly predicts per-frequency step-size numerators, error-coupling gates, and activity masks, replacing all heuristic update control with learned optimization. Experiments under matched settings show that the proposed system substantially outperforms a strong E2E cascade-solve baseline in perceptual speech quality (PESQ) and signal-to-distortion ratio (SI-SDR) during double-talk, at near-parity echo return loss enhancement (ERLE). Ablation shows this large PESQ gain at near-parity ERLE is the principled outcome of joint component-loss minimization—inverting the classical PESQ-vs-ERLE postfilter trade-off entrenched by pipeline-isolated tuning.