Predicting spatiotemporal fields governed by partial differential equations (PDEs) from sparse sensor data is a critical and long-standing challenge in science and engineering. Recent deep learning approaches, particularly neural operators, have shown considerable promise in solving PDEs. However, their performance degrades significantly in the demanding regime of extreme sparsity, characterized by spatial sensor coverage of less than 1% and limited temporal observations. To overcome this limitation, we propose a novel framework that decouples the task into two stages: spatial reconstruction and temporal extrapolation. In the first stage, rather than reconstructing the high-dimensional physical field directly, our model learns to reconstruct the complete latent features from sparse observations—features that would otherwise be extracted from a dense field. This process is stabilized by a Vector Quantization (VQ) bottleneck, which discretizes the latent space. In the second stage, a decoder-only Transformer performs temporal extrapolation by autoregressively predicting the future sequence of these discrete latent indices. This design inherently allows the model to generalize to new initial conditions and varying forecast horizons, akin to standard autoregressive models. We validate our framework on three challenging benchmarks, achieving state-of-the-art (SOTA) performance under severe sparsity constraints. Furthermore, we introduce a challenging benchmark dataset based on fire dynamics simulations. On this benchmark, our model successfully forecasts the field's evolution 30 frames into the future from a single timeframe with less than 0.1% spatial observations—a result that pushes well beyond the capabilities of existing methods.