The microscopic origins of quantum phase transitions (QPTs) in complex materials and the fate of the Fermi liquid state at quantum criticality are often debated. Quantum dots and mesoscopic circuits provide an experimental framework for realizing known quantum impurity Hamiltonians that can feature tunable second-order QPTs. Here we report on the theoretical and experimental investigation  of the quantum phase transitions occurring in a mesoscopic system, a quantum dot coupled to a metallic grain and to lead electrodes. We establish theoretically the complex phase diagram of this device through detailed numerical renormalization group calculations and resolve a former controversy: We show that, counter- intuitively, stable lines of non-Fermi liquid spin and charge two-channel Kondo states emerge and coexist with SU(4) physics in this simple device. We demonstrate experimentally in unprecedented detail, with support from numerical computations, a universal crossover from a quantum critical non-Fermi liquid behavior to distinct Fermi liquid ground states in a regime, where our device realizes a spin-1/2 impurity exchange-coupled equally to two independent electronic reservoirs. Arbitrarily small detuning of the exchange couplings results in conven- tional screening of the spin by the more strongly coupled channel for energies below a Fermi liquid scale T*. We extract a quadratic dependence of T* on gate voltage close to criticality and validate an asymptotically exact conformal field theory description of the universal crossover between strongly correlated non-Fermi liquid and Fermi liquid states.