  In this paper, a new technique for reconstructing and identifying
hadronically decaying $\tau^+\tau^-$ pairs with a large Lorentz boost, referred
to as the di-$\tau$ tagger, is developed and used for the first time in the
ATLAS experiment at the Large Hadron Collider. A benchmark di-$\tau$ tagging
selection is employed in the search for resonant Higgs boson pair production,
where one Higgs boson decays into a boosted $b\bar{b}$ pair and the other into
a boosted $\tau^+\tau^-$ pair, with two hadronically decaying $\tau$-leptons in
the final state. Using 139 fb$^{-1}$ of proton$-$proton collision data recorded
at a centre-of-mass energy of 13 TeV, the efficiency of the di-$\tau$ tagger is
determined and the background with quark- or gluon-initiated jets misidentified
as di-$\tau$ objects is estimated. The search for a heavy, narrow, scalar
resonance produced via gluon$-$gluon fusion and decaying into two Higgs bosons
is carried out in the mass range 1$-$3 TeV using the same dataset. No
deviations from the Standard Model predictions are observed, and 95%
confidence-level exclusion limits are set on this model.
