Trivially: $\mathbb{E}\left\{x^l\right\} = \cases{0 \qquad\qquad\qquad l\,\,\text{odd} \\ \sigma^l (l-1)!! \qquad l\,\,\text{even}}$
Is there a similar version for two Gaussian random variables, i.e.:
$$ \mathbb{E}\left\{ x_1^l x_2^l \right\} = \mathbb{E}\left\{ (x_1 x_2)^l \right\} $$
where $x_1$ and $x_2$ are Gaussian iid?
If $X_1$ and $X_2$ are independent then so are $X_1^n$ and $X_2^n$, hence $\mathbb{E}[X_1^nX_2^n]=\mathbb{E}[X_1^n]\mathbb{E}[X_2^n]$. This reduces the problem to the result you've stated.