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Twisted bilayer graphene develops quasiflat bands at specific "magic" interlayer rotation angles through an unconventional mechanism connected to carrier chirality. Quasiflat bands are responsible for a wealth of exotic, correlated-electron phases in the system. In this Letter, we propose a mechanical analog of twisted bilayer graphene made of two vibrating plates patterned with a honeycomb mesh of masses and coupled across a continuum elastic medium. We show that flexural waves in the device exhibit vanishing group velocity and quasiflat bands at magic angles in close correspondence with electrons in graphene models. The strong similarities of spectral structure and spatial eigenmodes in the two systems demonstrate the chiral nature of the mechanical flat bands. We derive analytical expressions that quantitatively connect the mechanical and electronic models, which allow us to predict the parameters required for an experimental realization of our proposal.