We present a design for a high-granularity calorimeter insert for future experiments at the Electron-Ion Collider (EIC). The sampling-calorimeter design uses scintillator tiles read out with silicon photomultipliers. It maximizes coverage close to the beampipe, while solving challenges arising from the beam-crossing angle and mechanical integration. It yields a compensated response that is linear over the energy range of interest for the EIC. Its energy resolution meets the requirements set in the EIC Yellow Report even with a basic reconstruction algorithm. Moreover, this detector will provide 5D shower data (position, energy, and time), which can be exploited with machine-learning techniques. This detector concept has the potential to unleash the power of imaging calorimetry at the EIC to enable measurements at extreme kinematics in electron–proton and electron–nucleus collisions.