You are a Senior Software Architect tasked with migrating a 10-year-old monolithic Java application to a microservices architecture. Define a strangler fig pattern strategy to incrementally extract services. Specifically, detail how you will handle shared database schema transitions to avoid dual-write consistency issues. Explain how you will implement an API Gateway to route traffic between the monolith and new services, and describe the circuit breaking and fallback mechanisms you will employ to ensure system stability during the transition phases.