  We report on Fe I in the day-side atmosphere of the ultra-hot Jupiter
WASP-33b, providing evidence for a thermal inversion in the presence of an
atomic species. We also introduce a new way to constrain the planet's
brightness variation throughout its orbit, including its day-night contrast and
peak phase offset, using high-resolution Doppler spectroscopy alone. We do so
by analyzing high-resolution optical spectra of six arcs of the planet's phase
curve, using ESPaDOnS on the Canada-France-Hawaii telescope and HDS on the
Subaru telescope. By employing a likelihood mapping technique, we explore the
marginalized distributions of parameterized atmospheric models, and detect Fe I
emission at high significance ($>10.4\sigma$) in our combined data sets,
located at $K_{\rm p}=222.1\pm0.4$ km/s and $v_{\rm sys}=-6.5\pm0.3$ km/s. Our
values agree with previous reports. By accounting for WASP-33b's brightness
variation, we find evidence that its night-side flux is $<10\%$ of the day-side
flux and the emission peak is shifted westward of the substellar point,
assuming the spectrum is dominated by Fe I. Our ESPaDOnS data, which cover
phases before and after the secondary eclipse more evenly, weakly constrain the
phase offset to $+22\pm12$ degrees. We caution that the derived
volume-mixing-ratio depends on our choice of temperature-pressure profile, but
note it does not significantly influence our constraints on day-night contrast
or phase offset. Finally, we use simulations to illustrate how observations
with increased phase coverage and higher signal-to-noise ratios can improve
these constraints, showcasing the expanding capabilities of high-resolution
Doppler spectroscopy.
