A search for the pair production of first-generation scalar leptoquarks is performed using proton-proton collision data recorded at 13 TeV center-of-mass energy with the CMS detector at the LHC. The data correspond to an integrated luminosity of 35.9 fb$^{-1}$. The leptoquarks are assumed to decay to a quark, and either an electron or a neutrino with branching fractions $\beta$ and 1$-\beta$, respectively. The search targets the decay final states comprising two electrons, or one electron and large missing transverse momentum, along with two quarks that are detected as hadronic jets. First-generation scalar leptoquarks with masses below 1435 (1270) GeV are excluded for $\beta =$ 1.0 (0.5). These are the most stringent limits on the mass of first-generation scalar leptoquarks to date. The data are also interpreted to set exclusion limits in the context of an $R$-parity violating supersymmetric model, predicting promptly decaying top squarks with a similar dielectron final state.