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AI-enabled precision medicine promises a transformational improvement in healthcare outcomes by enabling data-driven personalized diagnosis, prognosis, and treatment. However, the well-known "curse of dimensionality" and the clustered structure of biomedical data together interact to present a joint challenge in the high dimensional, limited observation precision medicine regime. To overcome both issues simultaneously we propose a simple and scalable approach to joint clustering and embedding that combines standard embedding methods with a convex clustering penalty in a modular way. This novel, cluster-aware embedding approach overcomes the complexity and limitations of current joint embedding and clustering methods, which we show with straightforward implementations of hierarchically clustered principal component analysis (PCA), locally linear embedding (LLE), and canonical correlation analysis (CCA). Through both numerical experiments and real-world examples, we demonstrate that our approach outperforms traditional and contemporary clustering methods on highly underdetermined problems (e.g., with just tens of observations) as well as on large sample datasets. Importantly, our approach does not require the user to choose the desired number of clusters, but instead yields interpretable dendrograms of hierarchically clustered embeddings. Thus our approach improves significantly on existing methods for identifying patient subgroups in multiomics and neuroimaging data, enabling scalable and interpretable biomarkers for precision medicine.