A data sample containing top quark pairs ($t\overline{t}$) produced in association with a Lorentz-boosted $Z$ or Higgs boson is used to search for signs of new physics using effective field theory. The data correspond to an integrated luminosity of $138\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of proton-proton collisions produced at a center-of-mass energy of 13 TeV at the LHC and collected by the CMS experiment. Selected events contain a single lepton and hadronic jets, including two identified with the decay of bottom quarks, plus an additional large-radius jet with high transverse momentum identified as a $Z$ or Higgs boson decaying to a bottom quark pair. Machine learning techniques are employed to discriminate between $t\overline{t}Z$ or $t\overline{t}H$ events and events from background processes, which are dominated by $t\overline{t}+\text{jets}$ production. No indications of new physics are observed. The signal strengths of boosted $t\overline{t}Z$ and $t\overline{t}H$ production are measured, and upper limits are placed on the $t\overline{t}Z$ and $t\overline{t}H$ differential cross sections as functions of the $Z$ or Higgs boson transverse momentum. The effects of new physics are probed using a framework in which the standard model is considered to be the low-energy effective field theory of a higher energy scale theory. Eight possible dimension-six operators are added to the standard model Lagrangian, and their corresponding coefficients are constrained via fits to the data.