Traffic congestion has significant economic, environmental, and social ramifications. Intersection traffic flow dynamics are influenced by numerous factors. While microscopic traffic simulators are valuable tools, they are computationally intensive and challenging to calibrate. Moreover, existing machine-learning approaches struggle to provide lane-specific waveforms or adapt to intersection topology and traffic patterns. In this study, we propose two efficient and accurate "Digital Twin" models for intersections, leveraging Graph Attention Neural Networks (GAT). These attentional graph auto-encoder digital twins capture temporal, spatial, and contextual aspects of traffic within intersections, incorporating various influential factors such as high-resolution loop detector waveforms, signal state records, driving behaviors, and turning-movement counts. Trained on diverse counterfactual scenarios across multiple intersections, our models generalize well, enabling the estimation of detailed traffic waveforms for any intersection approach and exit lanes. Multi-scale error metrics demonstrate that our models perform comparably to microsimulations. The primary application of our study lies in traffic signal optimization, a pivotal area in transportation systems research. These lightweight digital twins can seamlessly integrate into corridor and network signal timing optimization frameworks. Furthermore, our study's applications extend to lane reconfiguration, driving behavior analysis, and facilitating informed decisions regarding intersection safety and efficiency enhancements. A promising avenue for future research involves extending this approach to urban freeway corridors and integrating it with measures of effectiveness metrics.