What are some tricky ways to compute the conditional expectation other then using the conditional density?

Question

for one I know that if $X \perp \!\!\! \perp Y $ then for every measurable function $h$ we have

$$\mathbb{E}[h(X,Y) \mid X] = \phi(X)$$ where $\phi(x) =\mathbb{E}[h(x,Y)] $.

Also if you have a Gaussian vector $(X,Y)$ then $$\mathbb{E}[Y \mid X] = \mathbb{E}[Y ] + \frac{\operatorname{Cov}(X,Y)}{\operatorname{Var}X}[X - \mathbb{E}[ X]].$$

Any other trick or formula I should know ?