Single giant unilamellar vesicles (GUVs) rupture spontaneously from their salt-laden suspension onto solid surfaces. At hydrophobic surfaces, the GUVs rupture via a recurrent, bouncing ball rhythm. During each contact, the GUVs, rendered tense by the substrate interactions, porate, and spread a molecularly transformed motif of a monomolecular layer on the hydrophobic surface from the point of contact in a symmetric manner. The competition from pore closure, however, limits the spreading and produces a daughter vesicle, which re-engages with the substrate. At solid hydrophilic surfaces, by contrast, GUVs rupture via a distinctly different recurrent burst-heal dynamics; during burst, single pores nucleate at the contact boundary of the adhering vesicles, facilitating asymmetric spreading and producing a "heart"-shaped membrane patch. During the healing phase, the competing pore closure produces a daughter vesicle. In both cases, the pattern of burst-reseal events repeats multiple times, splashing and spreading the vesicular fragments as bilayer patches at the solid surface in a pulsatory manner. These remarkable recurrent dynamics arise, not because of the elastic properties of the solid surface, but because the competition between membrane spreading and pore healing, prompted by the surface-energy-dependent adhesion, determine the course of the topological transition.