Recent advances in machine learning and natural language processing have the potential to transform human activity in many domains. The field of learning analytics has applied these techniques successfully to many areas of education but has not been able to permeate others, such as doctoral education. Indeed, doctoral education remains an under-researched area with widespread problems (high dropout rates, low mental well-being) and lacks technological support beyond very specialized tasks. The inherent uniqueness of the doctoral journey may help explain the lack of generalized solutions (technological or otherwise) to these challenges. We propose a novel approach to apply the aforementioned advances in computation to support doctoral education. Single-case learning analytics defines a process in which doctoral students, researchers, and computational elements collaborate to extract insights about a single (doctoral) learner's experience and learning process. The feasibility and added value of this approach are demonstrated using an authentic dataset collected by nine doctoral students over a period of at least two months. The insights from this exploratory proof-of-concept serve to spark a research agenda for future technological support of doctoral education, which is aligned with recent calls for more human-centred approaches to designing and implementing learning analytics technologies.