Aims. We present a novel method for studying the thermal emission of exoplanets as a function of orbital phase at very high spectral resolution, and use it to investigate the climate of the ultra-hot Jupiter KELT-9b. Methods. We combine three nights of HARPS-N and two nights of CARMENES optical spectra, covering orbital phases between quadratures (0.25 < φ < 0.75), when the planet shows its day-side hemisphere with different geometries. We co-add the signal of thousands of Fe I lines through cross-correlation, which we map to a likelihood function. We investigate the phase-dependence of two separate observable quantities, namely (i) the line depths of Fe I and (ii) their Doppler shifts, introducing a new method that exploits the very high spectral resolution of our observations. Results. We confirm a previous detection of Fe I emission, and demonstrate a precision of 0.5 km s⁻¹ on the orbital properties of KELT-9b when combining all nights of observations. By studying the phase-resolved Doppler shift of Fe I lines, we detect an anomaly in the planet's orbital radial velocity well-fitted with a slightly eccentric orbital solution (e = 0.016 ± 0.003, ω = 150₋₁₁⁺¹³°, 5σ preference). However, we argue that this anomaly is caused by atmospheric circulation patterns, and can be explained if neutral iron gas is advected by day-to-night atmospheric wind flows of the order of a few km s⁻¹. We additionally show that the Fe I emission line depths are symmetric around the substellar point within 10° (2σ), possibly indicating the lack of a large hot-spot offset at the altitude probed by neutral iron emission lines. Finally, we do not obtain a significant preference for models with a strong phase-dependence of the Fe I emission line strength. We show that these results are qualitatively compatible with predictions from general circulation models (GCMs) for ultra-hot Jupiter planets. Conclusions. Very high-resolution spectroscopy phase curves are of sufficient sensitivity to reveal a phase dependence in both the line depths and their Doppler shifts throughout the orbit. They constitute an under-exploited treasure trove of information that is highly complementary to space-based phase curves obtained with HST and JWST, and open a new window onto the still poorly understood climate and atmospheric structure of the hottest planets known to date.