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Even though for generic $\mathcal{N}=1$ theories it is not possible toseparate distinct branches of supersymmetric vacua, in this paper we study aspecial class of $\mathcal{N}=1$ SCFTs, these of Class $\mathcal{S}_k$ forwhich it is possible to define Coulomb and Higgs branches precisely as for the$\mathcal{N}=2$ theories of Class $\mathcal{S}$ from which they descend. Westudy the BPS operators that parameterise these branches of vacua using thedifferent limits of the superconformal index as well as the Coulomb and Higgsbranch Hilbert Series. Finally, with the tools we have developed, we provide acheck that six dimensional $(1,1)$ Little String theory can be deconstructedfrom a toroidal quiver in class $\mathcal{S}_k$

The Coulomb and Higgs Branches of $\mathcal{N}=1$ Theories of Class $\mathcal{S}_k$