I'm not surprised you're having difficulty deriving the result; I think it's rather badly explained in the paper. Here's how I would describe the derivation:
In the paper, $a^e$ is introduced not as an equilibrium solution, but as "the public's perception of $a$" (p. 172, above (4)). In (3) and (10), the expectation values $E(\theta|x)$ and $E(\theta|y)$, respectively, refer to calculations performed by the public, whereas the expectation operator $\mathsf E$ (you didn't reproduce the typographical distinction made in the paper) refers to calculations performed by the bureaucrat (who can use the correct value $a$). The public decides about the career prospects of the bureaucrat in accordance with its estimation of the bureaucrat's talents, and it can only do so based on its perception $a^e$ of the bureaucrat's effort $a$ and on the policy outcome $y$. The paper does not specify how the public models the difference between its perception $a^e$ and the actual value $a$, but it seems from (4) that the public simply calculates as if the actual effort where known to be $a^e$. If we transfer that to the present setting including the unobservable noise $\varepsilon$, the public will use the model
where $\theta^e$ is the public's idea of the talent $\theta$, which it models according to the given normal distribution. Since $y=\theta+\varepsilon+a$, we have $\theta^e=\theta+a-a^e$. Now the public will apply your result, which I'll rewrite for $c$, $d$ and $x$ since the version with $a$, $b$ and $y$ is rather confusing (with $a$ referring to $\theta$ rather than $a$ in the present case): For $x=c+d$, with $c$ and $d$ i.i.d. normally distributed random variables with zero mean, $E[c|x]=(\sigma_c^2/\sigma_x^2)x$. Rewritten in terms of zero-mean variables, the public's model is
so we can apply the general result with $x=y-\bar\theta-a^e$, $c=\theta^e-\bar\theta$ and $d=\varepsilon$. Thus, the public's calculation of the expected talent of the bureaucrat given its perception $a^e$ of the effort and the observed policy outcome $y$ is
in agreement with the result in the paper.