  This paper is concerned with the nodal set of weak solutions to a broken
quasilinear partial differential equation, \begin{equation*} \mbox{div} (a_+
\nabla u^+ - a_- \nabla u^-) = \mbox{div} f, \end{equation*} where $a_+$ and
$a_-$ are uniformly elliptic, Dini continuous coefficient matrices, subject to
a strong correlation that $a_+$ and $a_-$ 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 H\"older
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.
