  We present observations of ASASSN-19dj, a nearby tidal disruption event (TDE)
discovered in the post-starburst galaxy KUG 0810+227 by the All-Sky Automated
Survey for Supernovae (ASAS-SN) at a distance of d $\simeq98$ Mpc. We observed
ASASSN-19dj from $-$21 to 392 d relative to peak ultraviolet (UV)/optical
emission using high-cadence, multiwavelength spectroscopy and photometry. From
the ASAS-SN $g$-band data, we determine that the TDE began to brighten on 2019
February 6.8 and for the first 16 d the rise was consistent with a flux
$\propto t^2$ power-law. ASASSN-19dj peaked in the UV/optical on 2019 March 6.5
(MJD = 58548.5) at a bolometric luminosity of $L = (6.2 \pm 0.2) \times 10^{44}
\text{ erg s}^{-1}$. Initially remaining roughly constant in X-rays and slowly
fading in the UV/optical, the X-ray flux increased by over an order of
magnitude $\sim$225 d after peak, resulting from the expansion of the X-ray
emitting region. The late-time X-ray emission is well fitted by a blackbody
with an effective radius of $\sim1 \times 10^{12} \text{ cm}$ and a temperature
of $\sim6 \times 10^{5} \text{ K}$. The X-ray hardness ratio becomes softer
after brightening and then returns to a harder state as the X-rays fade.
Analysis of Catalina Real-Time Transient Survey images reveals a nuclear
outburst roughly 14.5 yr earlier with a smooth decline and a luminosity of
$L_V\geq1.4 \times 10^{43}$ erg s$^{-1}$, although the nature of the flare is
unknown. ASASSN-19dj occurred in the most extreme post-starburst galaxy yet to
host a TDE, with Lick H$\delta_{A}$ = $7.67\pm0.17$ \AA.
