This paper introduces FoodSEM, a state-of-the-art fine-tuned open-source large language model (LLM) for named-entity linking (NEL) to food-related ontologies. To the best of our knowledge, food NEL is a task that cannot be accurately solved by state-of-the-art general-purpose (large) language models or custom domain-specific models/systems. Through an instruction-response (IR) scenario, FoodSEM links food-related entities mentioned in a text to several ontologies, including FoodOn, SNOMED-CT, and the Hansard taxonomy. The FoodSEM model achieves state-of-the-art performance compared to related models/systems, with F1 scores even reaching 98% on some ontologies and datasets. The presented comparative analyses against zero-shot, one-shot, and few-shot LLM prompting baselines further highlight FoodSEM's superior performance over its non-fine-tuned version. By making FoodSEM and its related resources publicly available, the main contributions of this article include (1) publishing a food-annotated corpora into an IR format suitable for LLM fine-tuning/evaluation, (2) publishing a robust model to advance the semantic understanding of text in the food domain, and (3) providing a strong baseline on food NEL for future benchmarking.