Time-dependent energy spectra of galactic cosmic rays (GCRs) carry crucial information regarding their origin and propagation throughout the interstellar environment. When observed at the Earth, after traversing the interplanetary medium, such spectra are heavily affected by the solar wind and the embedded solar magnetic field permeating the inner sectors of the heliosphere. The activity of the Sun changes significantly over an 11 yr solar cycle—and so does the effect on cosmic particles; this translates into a phenomenon called solar modulation. Moreover, GCR spectra during different epochs of solar activity provide invaluable information for a complete understanding of the plethora of mechanisms taking place in various layers of the Sun’s atmosphere and how they evolve over time. The High-Energy Particle Detector (HEPD-01) has been continuously collecting data since 2018 August, during the quiet phase between solar cycles 24 and 25; the activity of the Sun is slowly but steadily rising and is expected to peak around 2025/2026. In this paper, we present the first spectra for ∼50–250 MeV galactic protons measured by the HEPD-01 instrument—placed on board the CSES-01 satellite—from 2018 August to 2022 March over a one-Carrington-rotation time basis. Such data are compared to the ones from other spaceborne experiments, present (e.g., EPHIN, Parker Solar Probe) and past (PAMELA), and to a state-of-the-art three-dimensional model describing the GCRs propagation through the heliosphere.