Design-time safety guarantees for human-centered autonomous systems (HCAS) often break down in open-world deployment due to uncertain human interaction. In practice, HCAS must follow a user-personalized safety plan, with the human providing external inputs to handle out-of-distribution events. Open-world safety planning for HCAS demands modeling dynamical systems, exploring novel actions, and rapid replanning when plans are invalidated or dynamics shift. No single state-of-the-art planner meets all these needs. We introduce an LLM-based architecture that automatically generates personalized safety plans. By itself, the LLM fares poorly at producing safe usage plans, but coupling it with a safety verifier—which evaluates plan safety over the planning horizon and feeds back quality scores—enables the discovery of safe plans. Moreover, fine-tuning the LLM on personalized models inferred from open-world data further enhances plan quality. We validate our approach by generating safe usage plans for artificial pancreas systems in automated insulin delivery for Type 1 Diabetes patients. Code: https://github.com/ImpactLabASU/LLMOpen