The Laplacian matrix of a nondirected graph can be written as $M^TM$ with $M$ being the incidence matrix of the graph. This makes the (otherwise tedious) proof of Kirkhoff's theorem into a beautiful application of the Cauchy-Binet formula (and indeed, this is one of the proofs in "Proofs from THE BOOK").
If the graph is directed, $M^TM$ does not work anymore; the diagonal of the resulting matrix contains the total degree of vertices, whereas for Kirkhoff's theorem to work, only the indegree should appear.
So my question is this: can this approach still be salvaged by a slightly different definition of $M$ that eludes me, or is the "tedious" proof necessary and Cauchy-Binet simply can't be used here?