In this work, we demonstrate the growth of Ge crystals and suspended continuous layers on Si(001) substrates deeply patterned in high aspect-ratio pillars. The material deposition was carried out in a commercial reduced-pressure chemical vapor deposition reactor, thus extending the "vertical-heteroepitaxy" technique developed by using the peculiar low energy plasma-enhanced chemical vapor deposition reactor, to widely available epitaxial tools. The growth process was thoroughly analyzed, from the formation of small initial seeds to the final coalescence into a continuous suspended layer, by means of scanning and transmission electron microscopy, X-ray diffraction, and mu-Raman spectroscopy. The preoxidation of the Si pillar sidewalls and the addition of hydrochloric gas in the reactants proved to be key to achieve highly selective Ge growth on the pillars top only, which, in turn, is needed to promote the formation of a continuous Ge layer. Thanks to continuum growth models, we were able to single out the different roles played by thermodynamics and kinetics in the deposition dynamics. We believe that our findings will open the way to the low-cost realization of tens of micrometers thick heteroepitaxial layer (e.g., Ge, SiC, and GaAs) on Si having high crystal quality.