Large Language Models (LLMs) excel in understanding, generating, and processing human language, with growing adoption in process mining. Process mining relies on event logs that capture explicit process knowledge; however, knowledge-intensive processes (KIPs) in domains such as healthcare and product development depend on tacit knowledge, which is often absent from event logs. To bridge this gap, this study proposes a LLM-based framework for mobilizing tacit process knowledge and enriching event logs. A proof-of-concept is demonstrated using a KIP-specific LLM-driven conversational agent built on GPT-4o. The results indicate that LLMs can capture tacit process knowledge through targeted queries and systematically integrate it into event logs. This study presents a novel approach combining LLMs, knowledge management, and process mining, advancing the understanding and management of KIPs by enhancing knowledge accessibility and documentation.