Recent advancements in Large Language Models (LLMs) have primarily focused on enhancing task-specific performances by experimenting with prompt design. Despite the proven effectiveness of Metacognitive Prompting (MP), its application in the field of ontology generation remains an uncharted territory. This study addresses this gap by exploring this prompting technique in supporting the ontology design process, particularly with GPT-4, where this strategy has demonstrated consistent superiority over conventional and more direct prompting methods in recent research. Our methodology, named Ontogenia, employs a gold-standard dataset of ontology competency questions translated into SPARQL-OWL queries. This approach allows us to explore various types and stages of knowledge refinement using MP, while adhering to the eXtreme Design methodology, a well-established protocol in ontology design. Finally, the quality and performance of the resulting ontologies are assessed using both standard ontology quality metrics and evaluation by an ontology expert. This research aims to enrich the discussion on methods of ontology generation driven by LLMs by presenting concrete results on the use of metacognitive prompting and ontology design patterns.