Safety- and security-critical systems have to be thoroughly tested against their specifications. The state of practice is to have _natural language_ specifications, from which test cases are derived manually - a process that is slow, error-prone, and difficult to scale. _Formal_ specifications, on the other hand, are well-suited for automated test generation, but are tedious to write and maintain. In this work, we propose a two-stage pipeline that uses large language models (LLMs) to bridge the gap: First, we extract _protocol elements_ from natural-language specifications; second, leveraging a protocol implementation, we synthesize and refine a formal _protocol specification_ from these elements, which we can then use to massively test further implementations. We see this two-stage approach to be superior to end-to-end LLM-based test generation, as 1. it produces an _inspectable specification_ that preserves traceability to the original text; 2. the generation of actual test cases _no longer requires an LLM_; 3. the resulting formal specs are _human-readable_, and can be reviewed, version-controlled, and incrementally refined; and 4. over time, we can build a _corpus_ of natural-language-to-formal-specification mappings that can be used to further train and refine LLMs for more automatic translations. Our prototype, AUTOSPEC, successfully demonstrated the feasibility of our approach on five widely used _internet protocols_ (SMTP, POP3, IMAP, FTP, and ManageSieve) by applying its methods on their _RFC specifications_ written in natural-language, and the recent _I/O grammar_ formalism for protocol specification and fuzzing. In its evaluation, AUTOSPEC recovers on average 92.8% of client and 80.2% of server message types, and achieves 81.5% message acceptance across diverse, real-world systems.