consider the square $[-1,1]^2$ and a ball of radius $R$ entered at the origin $B_R(0)$. The function $u(x,y)=- \frac{\ln(\max(r^2,R^2))}{2}$ solves the Laplace problem $-\Delta u=0$, and the jump of the gradient is constant along $\partial B_R(0)$.
Goal: I'd like to find an easy solution where the gradient is not constant along $\partial B_R(0)$. Of course the rhs can be non-zero, as I am just building a manufactured solution. Can $u=- \frac{(x+y)\ln(\max(r^2,R^2))}{2}$ be a good candidate?
Complex analysis can help here. Say, gluing together on $\partial B_R(0)$ real parts of functions $z$ and $1/z$ inside and outside the circle gives $$ u(x,y)=\left\{ \begin{array}{c} x, r\le R,\\ \frac{R^2 x}{r^2}, r> R. \end{array} \right. $$