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 $simeq$ 98 Mpc. We observed ASASSN-19dj from $-$21 to 392 days relative to peak UV/optical emission using high-cadence, multi-wavelength 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 25 days 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 days after peak, resulting from the expansion of the X-ray emitting surface. The late-time X-ray emission is well-fit by a blackbody with an effective radius of $sim 1 times 10^{12} text{ cm}$ and a temperature of $sim 6 times 10^{5} text{ K}$. Analysis of Catalina Real-Time Transient Survey images reveals a nuclear outburst roughly 14.5 years earlier with a smooth decline and a luminosity of $L_V$ $geq$ $1.4 times 10^{43}$ 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$delta_{A}$ = $7.67 pm 0.17$ AA.
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