  The production of a Higgs boson in association with a top-antitop quark pair
($t\bar{t}H$) holds significant importance in directly probing the top-quark
Yukawa coupling, which is related to various fundamental questions in high
energy physics. This paper focuses on the calculation of two-loop amplitudes
for $t\bar t H$ production at hadron colliders in the high-energy boosted
limit. The calculation employs our recently developed mass-factorization
formula. To validate the accuracy of our approximate methods, we compare our
results for the one-loop amplitudes and the two-loop infrared poles with the
exact calculations. We then provide predictions for the finite parts of the
two-loop amplitudes. By combining the contributions from real emissions, our
results can be utilized to compute the next-to-next-to-leading order
differential cross sections for $t\bar t H$ production in the high-energy
boosted limit.
