  We identify a set of sufficient local conditions under which a significant
portion of a Radon measure $\mu$ on $\mathbb{R}^{n+1}$ with compact support can
be covered by an $n$-uniformly rectifiable set at the level of a ball $B\subset
\mathbb{R}^{n+1}$ such that $\mu(B)\approx r(B)^n$. This result involves a
flatness condition, formulated in terms of the so-called $\beta_1$-number of
$B$, and the $L^2(\mu|_B)$-boundedness, as well as a control on the mean
oscillation on the ball, of the operator \begin{equation} T_\mu f(x)=\int
\nabla_x\mathcal{E}(x,y)f(y)\,d\mu(y). \end{equation} Here
$\mathcal{E}(\cdot,\cdot)$ is the fundamental solution for a uniformly elliptic
operator in divergence form associated with an $(n+1)\times(n+1)$ matrix with
H\"older continuous coefficients. This generalizes a work by Girela-Sarri\'on
and Tolsa for the $n$-Riesz transform. The motivation for our result stems from
a two-phase problem for the elliptic harmonic measure.
