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Use of the serverless paradigm in cloud application development is growing rapidly, primarily driven by its promise to free developers from the responsibility of provisioning, operating, and scaling the underlying infrastructure. However, modern cloud-edge infrastructures are characterized by large numbers of disparate providers, constrained resource devices, platform heterogeneity, infrastructural dynamicity, and the need to orchestrate geographically distributed nodes and devices over public networks. This presents significant management complexity that must be addressed if serverless technologies are to be used in production systems. This position paper introduces COGNIT, a major new European initiative aiming to integrate AI technology into cloud-edge management systems to create a Cognitive Cloud reference framework and associated tools for serverless computing at the edge. COGNIT aims to: 1) support an innovative new serverless paradigm for edge application management and enhanced digital sovereignty for users and developers; 2) enable on-demand deployment of large-scale, highly distributed and self-adaptive serverless environments using existing cloud resources; 3) optimize data placement according to changes in energy efficiency heuristics and application demands and behavior; 4) enable secure and trusted execution of serverless runtimes. We identify and discuss seven research challenges related to the integration of serverless technologies with multi-provider Edge infrastructures and present our vision for how these challenges can be solved. We introduce a high-level view of our reference architecture for serverless cloud-edge continuum systems, and detail four motivating real-world use cases that will be used for validation, drawing from domains within Smart Cities, Agriculture and Environment, Energy, and Cybersecurity.