A search for pair production of scalar leptoquarks, each decaying into either an electron or a muon and a top quark, is presented. This is the first leptoquark search using ATLAS data to investigate top-philic cross-generational couplings that could provide explanations for recently observed anomalies in $B$ meson decays. This analysis targets high leptoquark masses which cause the decay products of each resultant top quark to be contained within a single high-$pT$ large-radius jet. The full Run 2 dataset is exploited, consisting of 139 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of data collected from proton-proton collisions at $s=13$ TeV from 2015 to 2018 with the ATLAS detector at the CERN Large Hadron Collider. In the absence of any significant deviation from the background expectation, lower limits on the leptoquark masses are set at 1480 GeV and 1470 GeV for the electron and muon channel, respectively.