We present the first measurement of the single-differential $\nu_e + \bar{\nu}_e$ charged-current inclusive cross sections on argon in electron or positron energy and in electron or positron scattering cosine over the full angular range. Data were collected using the MicroBooNE liquid argon time projection chamber located off-axis from the Fermilab Neutrinos at the Main Injector beam over an exposure of $2.0\times10^{20}$ protons on target. The signal definition includes a 60 MeV threshold on the $\nu_e$ or $\bar{\nu}_e$ energy and a 120 MeV threshold on the electron or positron energy. The measured total and differential cross sections are found to be in agreement with the GENIE, NuWro, and GiBUU neutrino generators.