Regular expression for a language which doesn't look regular
The trick is to write $L$ as a different language. Let's consider the following similar language
$$ L' = \{ xy \in \{0,1\}^* : \#_0(x) = \#_1(y) \}, $$
where $\#_0(x)$ is the number of 0's in $x$.

I claim that $L' = (0+1)^*$. Indeed, let $w = w_1\ldots w_n$ be some arbitrary word, and define $\delta_w(i) = \#_0(w_1\ldots w_i) - \#_1(w_{i+1}\ldots w_n)$. The following properties are not hard to check:


$\delta_w(0) \leq 0$.
$\delta_w(n) \geq 0$.
$\delta_w(i+1) = \delta_w(i) + 1$ for $0 \leq i \leq n-1$.


This shows that $\delta_w(i) = 0$ for some $0 \leq i \leq n$, and so $w \in L'$.

Your case is very similar - I'll leave you to figure out the details.