Large Language Models (LLMs) are rapidly becoming critical infrastructure for enterprise applications, driving unprecedented demand for GPU-based inference services. A key operational challenge arises from the two-phase nature of LLM inference: a compute-intensive \emph{prefill} phase that processes user input, followed by a memory-bound \emph{decode} phase that generates output tokens. When these phases share GPU resources, prefill tasks throttle the processing speed of concurrent decodes, creating state-dependent contention. This contention is further complicated by workload heterogeneity, as different applications exhibit vastly different input and output lengths. We develop a stochastic control framework for scheduling heterogeneous LLM workloads across large GPU clusters. We formulate LLM inference as a multiclass many-server queueing network with state-dependent service rates, grounded in empirical iteration-time measurements. We analyze the fluid approximation of this system and solve steady-state linear programs that characterize optimal resource allocation. We design gate-and-route policies that regulate prefill admission and decode routing, and prove that they are asymptotically optimal in the many-GPU limit under both bundled and separate token-pricing schemes. We further extend the framework to incorporate Service Level Indicators (SLIs) such as latency and fairness, providing a general approach to constrained scheduling. Numerical experiments calibrated to empirical iteration-time data demonstrate that our policies outperform standard serving heuristics.

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