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Exceptional Laguerre-type differential expressions make up an infinite classof Schr\"odinger operators having rational potentials and one limit-circleendpoint. In this manuscript, the spectrum of all self-adjoint extensions for ageneral exceptional Laguerre-type differential expression is given in terms ofthe Darboux transformations which relate the expression to the classicalLaguerre differential expression. The spectrum is extracted from an explicitWeyl $m$-function, up to a sign.  The construction relies primarily on two tools: boundary triples, whichparameterize the self-adjoint extensions and produce the Weyl $m$-functions,and manipulations of Maya diagrams and partitions, which classify the seedfunctions defining the relevant Darboux transforms. Several examples arepresented.

The Spectrum of Self-Adjoint Extensions associated with Exceptional Laguerre Differential Expressions