We propose an extension of the Sachdev-Ye-Kitaev (SYK) model that exhibits a quantum phase transition from the previously identified non-Fermi-liquid fixed point to a Fermi-liquid-like state, while still allowing an exact solution in a suitable large-N limit. The extended model involves coupling the interacting N-site SYK model to a new set of pN peripheral sites with only quadratic hopping terms between them. The conformal fixed point of the SYK model remains a stable low-energy phase below a critical ratio of peripheral sites p < p(c) (n) that depends on the fermion filling n. The scrambling dynamics throughout the non-Fermi-liquid (NFL) phase is characterized by a universal Lyapunov exponent lambda(L) -> 2 pi T in the low-temperature limit; however, the temperature scale marking the crossover to the conformal regime vanishes continuously at the critical point p(c). The residual entropy at T -> 0, nonzero in the NFL, also vanishes continuously at the critical point. For p > p(c) the quadratic sites effectively screen the SYK dynamics, leading to a quadratic fixed point in the low-temperature and low-frequency limit. The interactions have a perturbative effect in this regime leading to scrambling with Lyapunov exponent lambda(L) alpha T-2.