The emergence of 6G networks will pave the way for a diverse range of services to function within virtualized multi-cloud environments in the Edge-to-Cloud Continuum. This flexible and distributed architecture presents numerous prospects for enhancing service attributes such as availability, fault tolerance, and security. One primary instrument to this end is Moving Target Defense (MTD) as a security and resilience technique. However, MTD also poses an optimization challenge that extends beyond bolstering security alone and calls for smart and cognitive control. This paper elaborates on key technical topics regarding cognitive MTD in the 6G Edgeto- Cloud Continuum, including multi-objective modeling, RL-based control, and MTD peculiarities in 6G. We specifically highlight key technical challenges and identify potential directions for future research. As a concrete example, we present a Multi-objective Deep Reinforcement Learning (MORL) method to learn an optimal MTD strategy. Finally, we provide its preliminary performance evaluation in a 5G test bed embedding some key technologies applicable to future 6G networks, namely virtualized functions and AI-enhanced management and orchestration. The experimental results highlight the significance of MORL optimization over traditional deep-RL algorithms in this context.