The traditional generalized sidelobe canceller (GSC) is a common speech enhancement front end to improve the noise robustness of automatic speech recognition (ASR) systems in the far-field cases. However, the traditional GSC is optimized based on the signal level criteria, causing it not to guarantee the optimal ASR performance. To address this issue, we propose a novel dual-channel deep neural network (DNN)-based GSC structure, called nnGSC, which is optimized by using the objective of maximizing the ASR performance. Our key idea is to make each module of the traditional GSC fully learnable and use the acoustic model to perform joint optimization with GSC. We use the coefficients of the traditional GSC to initialize nnGSC, so that both traditional signal processing knowledge and large amounts of data can be used to guide the network learning. In addition, nnGSC can automatically track the target direction-of-arrival (DOA) frame-by-frame without the need for additional localization algorithms. In the experiments, nnGSC achieves a relative character error rate (CER) improvement of 23.7% compared to the microphone observation, 13.5% compared to the oracle direction-based super-directive beamformer, 12.2% compared to the oracle direction-based traditional GSC and 5.9% compared to the oracle mask-based minimum variance distortionless response (MVDR) beamformer. Moreover, we can improve the robustness of nnGSC against array geometry mismatches by training with multi-geometry data.