Osmotic deflation of vesicles enclosing two liquid phases can lead to bulging of one of the phases from the vesicle body. This budding process is preceded by a complete to partial wetting transition of one of the liquid phases on the membrane and depends on the membrane tensions and the tension of the interface between the enclosed liquid phases. These tensions dominate in different morphology regimes, the crossover of which initiates the budding process. In addition, the degree of budding can be controlled by aspiration via micropipets. We also demonstrate that the budding direction can be reversed if there are two external phases in contact with the vesicle.