The non-resonant production of a Higgs boson pair in association with a top-antitop quark pair \( \left( pp\to t\overline{t} hh\right) \) has only recently begun to be explored at the Large Hadron Collider (LHC) and provides a unique and largely uncharted probe of the top-Higgs sector, offering complementary sensitivity to the Higgs self-coupling and higher-dimensional interactions beyond the Standard Model. In this work, we present a detailed study of this process within the framework of Higgs Effective Field Theory (HEFT) at the High-Luminosity LHC (HL-LHC). A comparative analysis is performed using a traditional cut-based approach in the single-lepton channel and a multivariate parametric boosted decision tree method in both single-lepton and dilepton final states. We derive one- and two-parameter limits at 95% confidence level on the HEFT couplings δκλ, c2, c2g, ctg, and ctg2. The projected bound on δκλ is weaker than current experimental constraints from dedicated Higgs-pair measurement; however, this coupling plays a critical role in shaping the multidimensional allowed parameter space. For the remaining HEFT couplings, where no direct experimental limits currently exist, our results provide the first sensitivity projections in the \( t\overline{t} hh \) channel. Overall, this study demonstrates the strong potential of the \( t\overline{t} hh \) production process to probe extended Higgs and top-quark interactions beyond the Standard Model through the exploitation of the \( t\overline{t} hh \) data at the HL-LHC.
