Medical education resources are dense for common diseases but often sparse for under-covered conditions, atypical presentations, and fine-grained concept distinctions. This creates curriculum gaps that are difficult to repair manually, especially in long-tail domains where structured teaching materials are limited. We introduce Curriculum-Gap Completion (CGC), a new task for Large Language Model (LLM)-based medical education in which a model reconstructs missing educational units from a partially specified curriculum graph. Given topic nodes, pedagogical relations, and structured teaching slots, the model predicts omitted concepts, restores missing instructional links, and completes automatically verifiable teaching content. We instantiate this setting in a long-tail medical case study (hyperhidrosis) and evaluate five LLMs under three methods: direct prompting, retrieval-augmented prompting, and our proposed Structure-Aware Curriculum-Gap Completion (SACGC) framework. Across models, SACGC achieves the strongest overall performance, with the largest gains on structurally demanding masking settings. Ablation results show that explicit graph structure is the most important component, while schema constraints provide additional benefit. These findings suggest that LLMs are better suited for reconstructing an under-specified educational structure than for unrestricted medical tutoring, and they motivate CGC as a new natural language processing (NLP) problem for healthcare education.