Using inhomogeneous dynamical mean-field theory, we show that the normal-metal proximity effect could force any finite number of Mott-insulating barrier'' planes sandwiched between semi-infinite metallic leads to become fragile'' Fermi liquids. They are fully Fermi-liquid-like at T = 0, leading to a restoration of lattice periodicity at zero frequency, with a well-defined Fermi surface, and perfect (ballistic) conductivity. However, the Fermi-liquid character can rapidly disappear at finite omega, V, T, disorder, or magnetism, all of which restore the expected quantum tunneling regime, leading to fascinating possibilities for nonlinear response in devices.