We report the discovery of quasi-periodic oscillations (QPO) at ~11 mHz in two RXTE observations and one Chandra observation of the black hole candidate H1743-322. The QPO is observed only at the beginning of the 2010 and 2011 outbursts at similar hard colour and intensity, suggestive of an accretion state dependence for the QPO. Although its frequency appears to be correlated with X-ray intensity on timescales of a day, in successive outbursts eight months apart we measure a QPO frequency that differs by less than ~2.2 mHz while the intensity had changed significantly. We show that this ~11 mHz QPO is different from the so-called Type-C QPOs seen in black holes and that the mechanisms that produce the two flavours of variability are most probably independent. After comparing this QPO with other variability phenomena seen in accreting black holes and neutron stars, we conclude that it best resembles the so-called "1 Hz" QPOs seen in dipping neutron star systems, although having a significantly lower (1-2 orders of magnitude) frequency. If confirmed, H1743-322 is the first black hole showing this type of variability. Given the unusual characteristics and the hard-state dependence of the ~11 mHz QPO, we also speculate whether these oscillations could instead be related to the radio jets observed in H1743-322. A systematic search for this type of low-frequency QPOs in similar systems is needed to test this speculation. In any case, it remains unexplained why these QPOs have only been seen in the last two outbursts of H1743-322.