Clinical coding is a critical task in healthcare, although traditional methods for automating clinical coding may not provide sufficient explicit evidence for coders in production environments. This evidence is crucial, as medical coders have to make sure there exists at least one explicit passage in the input health record that justifies the attribution of a code. We therefore propose to reframe the task as an entity linking problem, in which each document is annotated with its set of codes and respective textual evidence, enabling better human-machine collaboration. By leveraging parameter-efficient fine-tuning of Large Language Models (LLMs), together with constrained decoding, we introduce three approaches to solve this problem that prove effective at disambiguating clinical mentions and that perform well in few-shot scenarios.