  Let $A(\cdot)$ be an $(n+1)\times (n+1)$ uniformly elliptic matrix with
H\"older continuous real coefficients and let $\mathcal E_A(x,y)$ be the
fundamental solution of the PDE $\mathrm{div} A(\cdot) \nabla u =0$ in $\mathbb
R^{n+1}$. Let $\mu$ be a compactly supported $n$-AD-regular measure in $\mathbb
R^{n+1}$ and consider the associated operator $$T_\mu f(x) = \int
\nabla_x\mathcal E_A(x,y)\,f(y)\,d\mu(y).$$ We show that if $T_\mu$ is bounded
in $L^2(\mu)$, then $\mu$ is uniformly $n$-rectifiable. This extends the
solution of the codimension $1$ David-Semmes problem for the Riesz transform to
the gradient of the single layer potential. Together with a previous result of
Conde-Alonso, Mourgoglou and Tolsa, this shows that, given $E\subset\mathbb
R^{n+1}$ with finite Hausdorff measure $\mathcal H^n$, if $T_{\mathcal H^n|_E}$
is bounded in $L^2(\mathcal H^n|_E)$, then $E$ is $n$-rectifiable. Further, as
an application we show that if the elliptic measure associated to the above PDE
is absolute continuous with respect to surface measure, then it must be
rectifiable, analogously to what happens with harmonic measure.
