  PSR J1809-1917 is a young ($\tau=51$ kyr) energetic
($\dot{E}=1.8\times10^{36}$ erg s$^{-1}$) radio pulsar powering an X-ray pulsar
wind nebula (PWN) that exhibits morphological variability. We report on the
results of a new monitoring campaign by the Chandra X-ray Observatory (CXO),
carried out across 6 epochs with a $\sim$7-week cadence. The compact nebula can
be interpreted as a jet-dominated outflow along the pulsar's spin axis. Its
variability can be the result of Doppler boosting in the kinked jet whose shape
changes with time (akin to the Vela pulsar jet). The deep X-ray image, composed
of 405 ks of new and 131 ks of archival CXO data, reveals an arcminute-scale
extended nebula (EN) whose axis of symmetry aligns with both the axis of the
compact nebula and the direction toward the peak of the nearby TeV source HESS
J1809-193. The EN's morphology and extent suggest that the pulsar is likely
moving through the ambient medium at a transonic velocity. We also resolved a
faint 7$'$-long nonthermal collimated structure protruding from the PWN. It is
possibly another instance of a "misaligned outflow" (also known as a "kinetic
jet") produced by high-energy particles escaping the PWN's confinement and
tracing the interstellar magnetic field lines. Finally, taking advantage of the
536 ks exposure, we analyzed the point sources in the J1809 field and
classified them using multiwavelength data. None of the classified sources in
the field can reasonably be expected to produce the extended TeV flux in the
region, suggesting that PSR J1809-1917 is indeed the counterpart to HESS/eHWC
J1809-193.
