While recent progresses in neural network approaches to single-channel speech separation, or more generally the cocktail party problem, achieved significant improvement, their performance for complex mixtures is still not satisfactory. In this work, we propose a novel multi-channel framework for multi-talker separation. In the proposed model, an input multi-channel mixture signal is firstly converted to a set of beamformed signals using fixed beam patterns. For this beamforming, we propose to use differential beamformers as they are more suitable for speech separation. Then each beamformed signal is fed into a single-channel anchored deep attractor network to generate separated signals. And the final separation is acquired by post selecting the separating output for each beams. To evaluate the proposed system, we create a challenging dataset comprising mixtures of 2, 3 or 4 speakers. Our results show that the proposed system largely improves the state of the art in speech separation, achieving 11.5 dB, 11.76 dB and 11.02 dB average signal-to-distortion ratio improvement for 4, 3 and 2 overlapped speaker mixtures, which is comparable to the performance of a minimum variance distortionless response beamformer that uses oracle location, source, and noise information. We also run speech recognition with a clean trained acoustic model on the separated speech, achieving relative word error rate (WER) reduction of 45.76\%, 59.40\% and 62.80\% on fully overlapped speech of 4, 3 and 2 speakers, respectively. With a far talk acoustic model, the WER is further reduced.