The ultra-hot Jupiter (UHJ) TOI-2109b marks the lower edge of the equilibrium temperature gap between 3500 and 4500 K, an unexplored thermal regime that separates KELT-9b, the hottest planet yet discovered, from all other currently known gas giants. To study the thermochemical structure of TOI-2109b's atmosphere, we obtained high-resolution emission spectra of both the planetary day- and nightsides with CAHA/CARMENES and VLT/CRIRES+. By applying the cross-correlation technique to the high-resolution spectra, we identified the emission signatures of Fe I (S/N = 4.3) and CO (S/N = 6.3), as well as a thermal inversion layer in the dayside atmo-sphere; no significant H2O signal was detected from the dayside. None of the analyzed species were detectable from the nightside atmosphere. We applied a Bayesian retrieval framework that combines high-resolution spectroscopy with photometric measurements to constrain the dayside atmospheric parameters and derive upper limits for the nightside hemisphere. The dayside thermal inversion extends from approximately 3200 to 4600 K, with an atmospheric metallicity consistent with that of the host star (0.36 dex). Only weak constraints could be placed on the C/O ratio, with a lower limit of 0.15. The retrieved spectral line broadening is consistent with tidally locked rotation, indicating the absence of strong dynamical processes in the atmosphere. An upper temperature limit of approximately 2400 K and a maximum atmospheric temperature gradient of about 700 K/log bar could be derived for the planetary nightside. Comparison of the retrieved dayside temperature-pressure profile with theoretical models, the absence of strong atmospheric dynamics, and significant differences in the thermal constraints between the day- and nightside hemispheres suggest a limited heat transport efficiency across the planetary atmosphere. Overall, our results place TOI-2109b in a transitional regime between the UHJs below the thermal gap, which show both CO and H2O emission lines, and KELT-9b, where molecular features are largely absent.