= Solution
By conformal invariance, $g_A(B)$ after its quadratic-variation time-change is Brownian motion in $\mathbb H$, started at
$$
g_A(iy)=x_y+i v_y,
\qquad x_y\to0,\quad v_y/y\to1.
$$
Since $E\subset\mathbb R\setminus[-1,1]$ and $A\subset\overline{\mathbb D}$, the boundary correspondence is regular there, and the exit event maps to $g_A(E)$. The <Poisson kernel> of $\mathbb H$ therefore gives
$$
\mathbb P^{iy}(B_{\tau_A}\in E)
=\int_{g_A(E)}
\frac{v_y}{(u-x_y)^2+v_y^2}\,\frac{du}{\pi}.
$$
For bounded subsets, multiplication by $y$ makes the integrand converge uniformly to $1/\pi$. Approximation by increasing bounded subsets and <monotone convergence theorem>[monotone convergence] then gives
$$
\lim_{y\to\infty}
y\,\mathbb P^{iy}(B_{\tau_A}\in E)
=\frac{\operatorname{Leb}(g_A(E))}{\pi},
$$
with both sides allowed to be infinite.
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