Mixture-of-Experts (MoE) architectures offer the promise of larger model capacity without the prohibitive costs of fully dense designs. However, in real-world inference serving, load skew across experts often leads to suboptimal device utilization and excessive synchronization overheads. This paper introduces Asynchronous Expert Parallelism (AEP), a new paradigm that decouples layer execution from barrier-style synchronization. By dynamically queuing tokens at each layer (referred to as $μ$-queuing) and adaptively re-batching them on demand, GPUs avoid waiting for straggling experts and instead continuously process whichever layer is ready. This asynchronous approach mitigates two major inefficiencies in traditional expert-parallel systems: (1) idle GPU time while waiting for the hottest expert, and (2) small-batch executions on colder experts that waste memory bandwidth. We implement these ideas in a serving system called AMoE, which disaggregates attention from expert layers and uses a defragging scheduler to reduce batch fragmentation. Evaluations on prototype MoE models show that AMoE improves throughput by up to 2.7x compared to state-of-the-art baselines, incurring a manageable latency penalty and providing a cost-effective operating point. Furthermore, experiments demonstrate nearly linear scalability to multi-node settings, whereas the baseline system shows no throughput increase even when the number of GPUs is doubled.