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As a generalization of Dillon's APN permutation, butterfly structure andgeneralizations have been of great interest since they generate permutationswith the best known differential and nonlinear properties over the field ofsize $2^{4k+2}$. Complementary to these results, we show in this paper thatbutterfly structure, more precisely the closed butterfly also yieldspermutations with the best boomerang uniformity, a new and important parameterrelated to boomerang-style attacks. This is the sixth known infinite family ofpermutations in the literature with the best known boomerang uniformity oversuch fields.

$4$-uniform BCT permutations from generalized butterfly structure