The Digital Transformation of our society requires IT infrastructures to be more agile, more adaptive, and more connected than ever. At the same time, the owners of such infrastructures are confronted with an increase in regulatory pressure (e.g., the GDPR). These developments put a lot of stress on IT management and governance. To enable IT management and governance to better deal with these challenges, we propose to digitally transform IT governance and management itself by using a Digital Twin based approach. In line with this, we aim to create on ontology-driven Digital Twin for Governed IT Management (DT4GITM) framework. The goal of this framework is to act as a reference architecture for a Digital Twin based infrastructure that connects three interrelated systems: the IT governance processes, the governed IT management processes, and the managed organizational IT assets. The core of the framework involves a generic Governed IT Management (GITM) Domain Ontology, which is planned to be operationalized by a Knowledge Graph based approach that realizes an integrated view on the heterogenous data streams originating from the IT governance and management processes, and the managed IT assets. In this paper, we start by outlining the planned DT4GITM framework and the pivotal role of the GITM Domain Ontology within this. We then elaborate our incremental, and scenario- and case-driven strategy towards the development of the framework as a whole, and the GITM Domain Ontology in particular. This is followed by the elaboration of a specific DT4GITM scenario which serves as a first proof of principle.