  PSR J1101-6101 is an energetic young pulsar which powers the remarkable
Lighthouse pulsar wind nebula (PWN). The pulsar belongs to the rare type of
radio- and gamma-ray-quiet pulsars which are bright in hard X-rays. Moreover,
the Lighthouse PWN is remarkable for its misaligned outflow (which gave rise to
the PWN's nickname). Also known as "pulsar filaments", these collimated
parsec-scale X-ray structures have been recently discovered in the vicinity of
a handful of fast-moving pulsars, and appear unaffected by the ram pressure
which confines pulsar tails. We report on NuSTAR observations of PSR J1101-6101
and its misaligned outflow -- the first observation of such a structure above
~10 keV. We detect the outflow up to 25 keV, spatially resolve its spectral
evolution with distance from the pulsar, find unambiguous evidence of spectral
cooling with distance from the pulsar, and infer physical properties of the
particles and magnetic field in the outflow. We also reanalzye archival Chandra
data and discuss the outflow's small-scale structure. We detect pulsations from
PSR J1101-6101 up to 20 keV, present the X-ray pulse profile, confirm its
period derivative, and perform phase-resolved spectroscopy. Lastly, we discuss
the X-ray source 2CXO J110158.4-605649 = 2XMM J110158.5--605651 (a
serendipitously observed blazar) and suggest it may be the X-ray counterpart to
the GeV source 4FGL J1102.0--6054.
