We present ${\rm poly\log\log n}$-round randomized distributed algorithms to compute vertex splittings, a partition of the vertices of a graph into $k$ parts such that a node of degree $d(u)$ has $\approx d(u)/k$ neighbors in each part. Our techniques can be seen as the first progress towards general ${\rm poly\log\log n}$-round algorithms for the Lovász Local Lemma. As the main application of our result, we obtain a randomized ${\rm poly\log\log n}$-round CONGEST algorithm for $(1+ε)Δ$-edge coloring $n$-node graphs of sufficiently large constant maximum degree $Δ$, for any $ε>0$. Further, our results improve the computation of defective colorings and certain tight list coloring problems. All the results improve the state-of-the-art round complexity exponentially, even in the LOCAL model.