Context. Classical Cepheids (DCEPs) are the most important primary indicators for the extragalactic distance scale, but they are also important objects in their own right, allowing us to place constraints on the physics of intermediate-mass stars and the pulsation theories. Aims. We have investigated the peculiar DCEP HD 344787, which is known to exhibit the fastest positive period change of DCEPs, along with a quenching amplitude of the light variation. Methods. We used high-resolution spectra obtained with HARPS-N at the TNG for HD 344787 and the more famous Polaris DCEP to infer their detailed chemical abundances. Results from the analysis of new time-series photometry of HD 344787 obtained by the TESS satellite are also reported. Results. The double-mode nature of the HD344787 pulsation is confirmed by an analysis of the TESS light curve, although with rather tiny amplitudes of a few dozen millimag. This is indication that HD344787 is on the verge of quenching the pulsation. Analysis of the spectra collected with HARPS-N at the TNG reveals an almost solar abundance and no depletion of carbon and oxygen. This means that the star appears to have not gone through first dredge-up. Similar results are obtained for Polaris. Conclusions. Polaris and HD344787 are both confirmed to be most likely at their first crossing of the instability strip. The two stars are likely at the opposite borders of the instability strip for first-overtone DCEPs with metal abundance Z = 0.008. A comparison with other DCEPs that are also thought to be at their first crossing allows us to speculate that the differences we see in the Hertzsprung-Russell diagram might be due to differences in the properties of the DCEP progenitors during the main-sequence phase.