The inclusive electron neutrino charged-current cross section is measured in the NOvA near detector using $8.02\times {10}^{20}$ protons-on-target in the NuMI beam. The sample of GeV electron neutrino interactions is the largest analyzed to date and is limited by $\simeq 17\%$ systematic rather than the $\simeq 7.4\%$ statistical uncertainties. The double-differential cross section in final-state electron energy and angle is presented for the first time, together with the single-differential dependence on $Q^2$ (squared four-momentum transfer) and energy, in the range $1\mathrm{GeV}\le E_{\nu }<6\mathrm{GeV}$. Detailed comparisons are made to the predictions of the GENIE, GiBUU, NEUT, and NuWro neutrino event generators. The data do not strongly favor a model over the others consistently across all three cross sections measured, though some models have especially good or poor agreement in the single differential cross section vs $Q^2$.