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The recognition in the 1990s that vortex beams (VBs), paraxial light beams with optical vortices, carry orbital angular momentum (OAM), has benefited applications ranging from optical manipulation to high-dimensional classical and quantum information communications. The transverse profiles of common VBs, e.g., Laguerre-Gaussian beam and high-order Bessel beam, are hollow donuts whose radii grow up with OAM inevitably. The inherently unperfect character of the VBs that the radius is always positively correlated with OAM, restricts the application of the VBs in many scenarios like fiber optic data transmission, spatial OAM mode (de)multiplexing communication, and particle manipulation, which call for VBs to have the same scale with distinct OAM or even the small vortex ring for large OAM. Here, we derived a theory based on the most widely used Laguerre-Gaussian beam to generate a brand new type of VB with OAM-independent radii that moves away from the common unperfect constraint, called Laguerre-Gaussian Perfect Vortex Beam (LGPVB). LGPVBs have the self-similar property like common Laguerre-Gaussian beams but can self-heal after suffering disturbance and always remain 'perfection' when propagating. Our Fourier-space design not only allows us to shape the LGPVB's propagating intensity at will, but it also gives LGPVB the fascinating potential to arbitrarily self-accelerate while still perfectly propagating, self-similar, and self-healing. This customizable self-healing LGPVB, whose properties inform our most expectations of VBs, offers a better alternative for application scenarios of common VBs in a wide range of areas.