A quasilinear elliptic equation of second order can be split into a first order system in various ways. We present and analyze a stabilized finite element method for the system, which is well suited for any of these possible splittings. Under minimal assumptions on the continuous solution, existence and (nearly) optimal convergence in $L^\infty$ of the discrete solutions is established. This result holds for any choice of the stabilization parameter $\omega>0$. Moreover, the paper presents a framework for investigating other mixed methods for unsymmetric first order systems.