The power density and, consequently, power hungriness of server processors is growing by the day. Traditional air cooling systems fail to cope with such high heat densities, whereas single-phase liquid-cooling still requires high mass flowrate, high pumping power, and large facility size. On the contrary, in a micro-scale gravity-driven thermosyphon attached on top of a processor, the refrigerant, absorbing the heat, turns into a two-phase mixture. The vapor-liquid mixture exchanges heat with a coolant at the condenser side, turns back to liquid state, and descends thanks to gravity, eliminating the need for pumping power. However, similar to other cooling technologies, thermosyphon efficiency can considerably vary with respect to workload performance requirements and thermal profile, in addition to the platform features, such as packaging and die floorplan. In this work, we first address the workload- and platform-aware design of a two-phase thermosyphon. Then, we propose a thermal-aware workload mapping strategy considering the potential and limitations of a two-phase thermosyphon to further minimize hot spots and spatial thermal gradients. Our experiments, performed on an 8-core Intel Xeon E5 CPU reveal, on average, up to 10ºC reduction in thermal hot spots, and 45% reduction in the maximum spatial thermal gradient on the die. Moreover, our design and mapping strategy are able to decrease the chiller cooling power at least by 45%.