This paper is concerned with the nodal set of weak solutions to a broken quasilinear partial differential equation, (Formula presented.) where (Formula presented.) and (Formula presented.) are uniformly elliptic, Dini continuous coefficient matrices, subject to a strong correlation that (Formula presented.) and (Formula presented.) are a multiple of some scalar function to each other. Under such a structural condition, we develop an iteration argument to achieve higher-order approximation of solutions at a singular point, which is also new for standard elliptic PDEs below Hölder regime, and as a result, we establish a structure theorem for singular sets. We also estimate the Hausdorff measure of nodal sets, provided that the vanishing order of given solution is bounded throughout its nodal set, via an approach that extends the classical argument to certain solutions with discontinuous gradient. Besides, we also prove Lipschitz regularity of solutions and continuous differentiability of their nodal set around regular points.