Small-scale jet-like eruptions, such as picoflare jets and jetlets, are recognized as potential contributors to coronal heating and solar wind acceleration, yet their physical origin is still not fully established. Using ultra-high-resolution extreme ultraviolet imaging datasets from the Extreme Ultraviolet Imager on board the Solar Orbiter mission, we investigate tiny coronal jets observed off-limb in the Sun's polar regions. Visual inspection reveals that the majority of these jets exhibit distinct morphological features, including a bright spire accompanied by a dark, eruptive jet component. We analyzed 11 of these jets in detail and found that their spatial and temporal scales are comparable to previously reported jetlets, while their kinetic energies are two to three orders of magnitude lower, placing them in the picoflare regime. The bright and dark components show distinct dynamics, with the dark structures generally displaying lower speeds. A comparison with coordinated Interface Region Imaging Spectrograph data and the Atmospheric Imaging Assembly on board the Solar Dynamics Observatory data, together with 2.5D radiative-MHD simulations performed with the Bifrost code, reveals a one-to-one morphological correspondence between the dark counterparts and cool chromospheric surges accompanying the bright jet spire. This association suggests that flux emergence and magnetic reconnection at low atmospheric heights may produce coupled bright─dark structures, providing a plausible mechanism for the generation of picoflare jets. Our results demonstrate Solar Orbiter's ability to resolve the dynamics of small-scale jets and place new constraints on their origin.