The Lovász Local Lemma is a classic result in probability theory that is often used to prove the existence of combinatorial objects via the probabilistic method. In its simplest form, it states that if we have $n$ `bad events', each of which occurs with probability at most $p$ and is independent of all but $d$ other events, then under certain criteria on $p$ and $d$, all of the bad events can be avoided with positive probability. While the original proof was existential, there has been much study on the algorithmic Lovász Local Lemma: that is, designing an algorithm which finds an assignment of the underlying random variables such that all the bad events are indeed avoided. Notably, the celebrated result of Moser and Tardos [JACM '10] also implied an efficient distributed algorithm for the problem, running in $O(\log^2 n)$ rounds. For instances with low $d$, this was improved to $O(d^2+\log^{O(1)}\log n)$ by Fischer and Ghaffari [DISC '17], a result that has proven highly important in distributed complexity theory (Chang and Pettie [SICOMP '19]). We give an improved algorithm for the Lovász Local Lemma, providing a trade-off between the strength of the criterion relating $p$ and $d$, and the distributed round complexity. In particular, in the same regime as Fischer and Ghaffari's algorithm, we improve the round complexity to $O(\frac{d}{\log d}+\log^{O(1)}\log n)$. At the other end of the trade-off, we obtain a $\log^{O(1)}\log n$ round complexity for a substantially wider regime than previously known. As our main application, we also give the first $\log^{O(1)}\log n$-round distributed algorithm for the problem of $Δ+o(Δ)$-edge coloring a graph of maximum degree $Δ$. This is an almost exponential improvement over previous results: no prior $\log^{o(1)} n$-round algorithm was known even for $2Δ-2$-edge coloring.