Let $X \sim \mathcal{N}(\mu, \sigma^2)$, $Y \sim U(-\alpha, \alpha)$ and $Z = X - Y$. Assume $Y$ and $Z$ are independent. Note that $X$ and $Y$ are necessarily dependent.
What is $\mathbb{E}[Z|X]$?
What I've tried
$$ \mathbb{E}[Z|X](x) = \frac{1}{p_X(x)} \int z \, p_Y(x-z)p_Z(z) \mathrm{d}z $$
I don't know what $p_Z(z)$ is. I've tried getting it from the characteristic functions $$ p_Z(z) = \frac{1}{2\pi}\int e^{-itz} \Phi_X(t)/\Phi_Y(t) \mathrm{d} t $$ but the integral looks intimidating.
It would be nicer to beat the integral for $\mathbb{E}[Z|X]$ out of $\Phi_Z(t)$ without finding a closed form for $p_Z(z)$...
The setup is not possible
We have $$X = Y+Z$$
hence
$$\Phi_X(t) = \Phi_Y(t) \Phi_Z(t) $$
Because the first factor is a $sinc$ function, the product cannot be a gaussian.