  4FGL J1015.5-6030 is an unidentified Fermi-LAT source hosting a bright,
extended X-ray source whose X-ray spectrum is consistent with that of a young
pulsar, yet no pulsations have been found. Here we report on XMM-Newton timing
and Chandra imaging observations of the X-ray counterpart of 4FGL J1015.5-6030.
We find no significant periodicity from the source and place a 3$\sigma$
upper-limit on its pulsed fraction of 34$\%$. The Chandra observations resolve
the point source from the extended emission. We find that the point source's
spectrum is well fit by a blackbody model, with temperature $kT=0.205\pm0.009$
keV, plus a weak power-law component, which is consistent with a thermally
emitting neutron star with a magnetospheric component. The extended emission
spans angular scales of a few arcseconds up to about 30$''$ from the point
source and its spectrum is well fit by a power-law model with a photon index
$\Gamma=1.70\pm0.05$. The extended emission's spectrum and 0.5-10 keV
luminosity of 4$\times10^{32}$ erg s$^{-1}$ (at a plausible distance of 2 kpc)
are consistent with that of a pulsar wind nebula. Based on a comparison to
other GeV and X-ray pulsars, we find that this putative pulsar is likely a
middle-aged (i.e., $\tau\sim 0.1$--1 Myr) radio-quiet pulsar with
$\dot{E}\sim10^{34}-10^{35}$ erg s$^{-1}$.
