High-fidelity thermal simulations are important for understanding and optimizing laser-based manufac- turing processes such as laser beam welding and Directed Energy Deposition (DED), yet traditional finite element methods remain too slow for multi-query applications such as process optimization and real-time control [1]. This work introduces a Neural Operator surrogate [2] that significantly reduces simulation time while maintaining predictive accuracy. The surrogate operates in a moving reference frame tracking the laser heat source, efficiently resolving localized thermal gradients while decoupling the computational cost from the global domain size. The ar- chitecture performs autoregressive predictions conditioned on physics-based inputs and geometric priors to generalize across varying process parameters and geometries. To address the fundamental challenges of autoregressive instability and limited spatial context in moving-window operators, we investigate sev- eral architectural and training strategies. A latent geometric context encoder processes extended-domain geometry at coarse resolution and fuses it with local thermal states via cross-attention, providing informa- tion of approaching boundaries at minimal computational cost. Additionally, we implement curriculum learning strategies that progressively increase rollout horizons and geometric complexity during training, improving stability and sample efficiency. Validation on various test scenarios generated by high-fidelity solvers [3] demonstrate robust autoregres- sive stability and significant speedups. This framework offers a scalable pathway toward high-fidelity Digital Twins, bridging the gap between accurate physics-based simulation and the real-time require- ments of advanced manufacturing systems.