Within the νMSSM, a Minimal Supersymmetric neutrino See-saw Model, Lepton Flavour Violating Higgs couplings are strongly enhanced at large tan β ( > ∼ 30), which can lead to BR(H 0 /A 0 → τ µ) ≃ O(10 −4 ), for M H 0 /A 0 > ∼ 160 GeV. Enhancements on the production of Higgs bosons, through the gluon fusion mechanism, gg → H 0 /A 0 , and the associated production channel gg, qq → bbH 0 /A 0 , whose rates grow with tan β, as well as the mass degeneracy that occurs between the H 0 and A 0 states in this regime, also contribute to further the possibilities to detect a heavy Higgs signal into τ µ pairs. We show that the separation of τ µ Higgs events from the background at the upcoming CERN Large Hadron Collider could be done for Higgs masses up to about 600 GeV for 300 fb −1 of luminosity, for large tan β values. However, even with as little as 10 fb −1 one can probe H 0 /A 0 masses up to 400 GeV or so, if tan β = 60. Altogether, these processes then provide a new Higgs discovery mode as well as an independent test of flavour physics.