In this letter we discuss the technique of graded-thickness sample deposition for studying the growth mechanisms of GaAs heteroepitaxy on Si. We can observe the continuous evolution from the initial clean surface, through nucleation, growth, and coalescence of the deposited material. We describe results for a sample typical of buffer layer growth in the two-step molecular beam epitaxial deposition of GaAs directly on Si. We are led to a specific model for the three-dimensional nucleation and growth mechanisms in which Ga atom diffiusion dominates the stable cluster formation, As capture from the vapor by the islands immobilizes the Ga, and island growth is limited by the binding of diffusing Ga.