Biometric systems aim to identify or verify individuals using physiological or behavioral traits reliably, but traditional modalities such as fingerprints, faces, and voices are vulnerable to spoofing and replay attacks. EEG has therefore emerged as a promising biometric alternative due to its inherent security, subject specificity, and resistance to forgery. However, a key limitation of existing EEG based biometric systems is poor cross session generalization, resulting in significant performance drops when models are tested on data from unseen recording sessions, which restricts real world deployment. In this work, we present a cross session EEG biometric identification framework evaluated on the publicly available BED dataset under eyes closed resting state conditions, using a strict protocol in which sessions one and two are used for training and validation, and session three is reserved exclusively for testing. We have proposed two hybrid CNN, LSTM, and encoder architectures, one with a four layer CNN paired with two layer stacked LSTMs and an encoder, and a deeper 8 layer CNN paired with four layer stacked LSTMs and an encoder. The cross session test performance for the four layer CNN and two layer LSTM pipeline achieved the highest test accuracy of 93.88% for six seconds, and the eight layer CNN and four layer LSTM pipeline reached a peak accuracy of 97.89% at two seconds. Finally, when compared with prior BED based cross session work that typically reports only a mid 80% cross session accuracies for CNN and classical pipelines, the proposed CNN LSTM and encoder design set a much higher accuracy mark on eyes closed BED Dataset while following to a strict cross session split and transparent extraction procedure with the highest accuracy of 97.89% for two seconds time segment the shortest time window we have experimented on in our work.