Lovász proves in one of his exercises from Combinatorial Problems and Exercises (1992) that if $G$ is a critical graph with $\chi(G) = k + 1$ then $G$ is $k-1$ edge connected.
In my own words I understand his proof to be as follows:
Assume that $m \leq k - 1$ edges $e_1, ... , e_m$ partition $G$ into 2 subgraphs $G_1$ and $G_2$ which are $k$ colorable by criticality of $G$; that is, assume that $G$ becomes disconnected with the removal of $\leq k - 1$ edges. Then, these graphs can be joined by $k - 1$ edges s.t. $G$ has a coloring in $k$ colors, a contradition. Thus, $G$ is a critical graph with $\chi(G) = k$ $\implies$ $G$ is $k-1$ edge connected. $\square$
Is what I have sufficient to prove the statement? It seems Lovász makes a more rigorous argument about why the union of $G_1$ and $G_2$ is colorable in $k$ colors in his original proof.