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In this work, we develop a two time-scale deep learning approach for beamforming and phase shift (BF-PS) design in time-varying RIS-aided networks. In contrast to most existing works that assume perfect CSI for BF-PS design, we take into account the cost of channel estimation and utilize Long Short-Term Memory (LSTM) networks to design BF-PS from limited samples of estimated channel CSI. An LSTM channel extrapolator is designed first to generate high resolution estimates of the cascaded BS-RIS-user channel from sampled signals acquired at a slow time scale. Subsequently, the outputs of the channel extrapolator are fed into an LSTM-based two stage neural network for the joint design of BF-PS at a fast time scale of per coherence time. To address the critical issue that training overhead increases linearly with the number of RIS elements, we consider various pilot structures and sampling patterns in time and space to evaluate the efficiency and sum-rate performance of the proposed two time-scale design. Our results show that the proposed two time-scale design can achieve good spectral efficiency when taking into account the pilot overhead required for training. The proposed design also outperforms a direct BF-PS design that does not employ a channel extrapolator. These demonstrate the feasibility of applying RIS in time-varying channels with reasonable pilot overhead.