Serving large generative models such as LLMs and multi- modal transformers requires balancing user-facing SLOs (e.g., time-to-first-token, time-between-tokens) with provider goals of efficiency and cost reduction. Existing solutions rely on static provisioning or model-level autoscaling, both of which treat the model as a monolith. This coarse-grained resource management leads to degraded performance or significant resource underutilization due to poor adaptability to dynamic inference traffic that is common online. The root cause of this inefficiency lies in the internal structure of generative models: they are executed as graphs of interconnected operators. Through detailed characterization and systematic analysis, we find that operators are heterogeneous in their compute and memory footprints and exhibit diverse sensitivity to workload and resource factors such as batch size, sequence length, and traffic rate. This heterogeneity suggests that the operator, rather than the entire model, is the right granularity for scaling decisions. We propose an operator-level autoscaling framework, which allocates resources at finer (operator)-granularity, optimizing the scaling, batching, and placement based on individual operator profiles. Evaluated on production-scale traces, our approach preserves SLOs with up to 40% fewer GPUs and 35% less energy, or under fixed resources achieves 1.6x higher throughput with 5% less energy. These results show that the operator, rather than the model, is fundamentally a more effective unit for scaling large generative workloads.