  [Abridged] We present a novel method to study the thermal emission of
exoplanets as a function of orbital phase at very high spectral resolution, and
apply it to investigate the climate of the ultra-hot Jupiter KELT-9b. We
combine 3 nights of HARPS-N and 2 nights of CARMENES optical spectra, covering
orbital phases between quadratures (0.25 < phi < 0.75), when the planet shows
its day-side hemisphere with different geometries. We co-add the signal of
thousands of FeI lines through cross-correlation, which we map to a likelihood
function. We investigate the phase-dependence of: (i) the line depths of FeI,
and (ii) their Doppler shifts, by introducing a new method that exploits the
very high spectral resolution of our observations. We confirm a previous
detection of FeI emission and demonstrate a combined precision of 0.5 km s-1 on
the orbital properties of KELT-9b. By studying the phase-resolved Doppler shift
of FeI lines, we detect an anomaly in the planet's orbital radial velocity
well-fitted with a slightly eccentric orbit (e = 0.016$\pm$0.003, w =
150$^{+13\circ}_{-11},~5\sigma$ preference). However, we argue that such
anomaly can be explained by a day-night wind of a few km s-1 blowing neutral
iron gas. Additionally, we find that the FeI emission line depths are symmetric
around the substellar point within 10 deg ($2\sigma$). We show that these
results are qualitatively compatible with predictions from general circulation
models for ultra-hot Jupiter planets. Very high-resolution spectroscopy phase
curves have the sensitivity to reveal a phase dependence in both the line
depths and their Doppler shifts throughout the orbit. They are highly
complementary to space-based phase curves obtained with HST and JWST, and open
a new window into the still poorly understood climate and atmospheric structure
of the hottest planets known.
