Query rewriting is a crucial task for improving retrieval, especially in professional domains such as law and medicine, where user queries are often underspecified and ambiguous. While large language models (LLMs) offer strong understanding and generation capabilities, existing LLM-based approaches reduce the task to text transformation or expansion, neglecting reasoning to disambiguate queries, which fails to bridge the cognitive gap between user queries and specialized documents. In this paper, we propose Think-Then-Rewrite (TTR), a reinforcement learning based framework that unleashes LLMs' reasoning ability for domain-specific query rewriting. TTR introduces a contrastive mutual information reward to encourage the LLM to generate reasoning processes that effectively distinguish confusing distractors. To boost early-stage training, TTR also constructs golden query rewrites as off‑policy data, providing strong guidance for RL learning. A mixed-policy optimization then combines on-policy and off-policy signals, ensuring both effectiveness and stability. Extensive experiments on legal and medical retrieval benchmarks demonstrate that TTR achieves state-of-the-art performance.