Measurements are made of differential cross-sections of highly boosted pair-produced top quarks as a function of top-quark and $t\overline{t}$ system kinematic observables using proton-proton collisions at a center-of-mass energy of $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$. The data set corresponds to an integrated luminosity of $36.1\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$, recorded in 2015 and 2016 with the ATLAS detector at the CERN Large Hadron Collider. Events with two large-radius jets in the final state, one with transverse momentum ${p}_{\mathrm{T}}>500\text{ }\text{ }\mathrm{GeV}$ and a second with ${p}_{\mathrm{T}}>350\text{ }\text{ }\mathrm{GeV}$, are used for the measurement. The top-quark candidates are separated from the multijet background using jet substructure information and association with a $b$-tagged jet. The measured spectra are corrected for detector effects to a particle-level fiducial phase space and a parton-level limited phase space, and are compared to several Monte Carlo simulations by means of calculated ${\ensuremath{\chi}}^{2}$ values. The cross-section for $t\overline{t}$ production in the fiducial phase-space region is $292\ifmmode\pm\else\textpm\fi{}7(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}71(\mathrm{syst})\text{ }\text{ }\mathrm{fb}$, to be compared to the theoretical prediction of $384\ifmmode\pm\else\textpm\fi{}36\text{ }\text{ }\mathrm{fb}$.