End-to-end automatic speech recognition (ASR) systems frequently misrecognize domain-specific phrases like named entities, which can cause catastrophic failures in downstream tasks. A new family of named entity correction methods based on large language models (LLMs) has recently emerged. However, these approaches have yet to fully exploit the sophisticated reasoning capabilities inherent to LLMs. To bridge this gap, we propose a novel retrieval-augmented generation framework for correcting named entity errors in ASR. Our approach consists of two key components: (1) a rephrasing language model (RLM) for named entity recognition, followed by candidate retrieval using a phonetic-level edit distance; and (2) a novel self-taught reasoning model with adaptive chain-of-thought (A-STAR) that dynamically adjusts the depth of its reasoning based on task difficulty. Experiments on the AISHELL-1 and Homophone datasets demonstrate the effectiveness of our method, which achieves relative reductions in the named entity character error rate of 17.96\% and 34.42\%, respectively, compared to a strong baseline.