Studies have linked e-government failures (primarily its adoption and development) to factors such as design shortcomings and design-reality gaps. With the increased availability of big data and analytical techniques, government agencies must leverage data-driven models in the design and implementation of electronic government (e-government) or digital government solutions. By doing so, minimizing design-reality gap problems and design shortcomings is attainable. Backed by stakeholder theory and the Agile design methodology, this study proposes a cognitive computing framework as an efficient approach to the design of public sector electronic services (e-services) so as to obtain value. Cognitive computing was deemed appropriate for inclusion into the Agile methodology for e-government design because of its rapid frequency of expansion of data for which the human mind is limited in analytical ability as well as the desire by policy makers to reduce burdens and cost against the desire for more tailored solutions. With respect to research, cognitive computing models present a novel means of implementing human-centric and data-driven processes into practical domains of discourse and these models have yet to be integrated into the e-government sphere. By applying design thinking, the proposed model is built and tested in the UPPAAL model-checking software by applying the principle of finite state deterministic automatons. Theoretical and practical contributions are made to the fields of e-government, Agile methodology, and quality service delivery. Future research recommendations for expansion and building evaluation metrics are also indicated in the study.