In practice, requirements specification remains a critical challenge. The knowledge necessary to generate a specification can often be fragmented across diverse sources (e.g., meeting minutes, emails, and high-level product descriptions), making the process cumbersome and time-consuming. In this paper, we report our experience using large language models (LLMs) in an IT consulting company to automate the requirements specification process. In this company, requirements are specified using a Functional Design Specification (FDS), a document that outlines the functional requirements and features of a system, application, or process. We provide LLMs with a summary of the requirements elicitation documents and FDS templates, prompting them to generate Epic FDS (including high-level product descriptions) and user stories, which are subsequently compiled into a complete FDS document. We compared the correctness and quality of the FDS generated by three state-of-the-art LLMs against those produced by human analysts. Our results show that LLMs can help automate and standardize the requirements specification, reducing time and human effort. However, the quality of LLM-generated FDS highly depends on inputs and often requires human revision. Thus, we advocate for a synergistic approach in which an LLM serves as an effective drafting tool while human analysts provide the critical contextual and technical oversight necessary for high-quality RE documentation.