  Exceptional Laguerre-type differential expressions make up an infinite class
of Schr\"odinger operators having rational potentials and one limit-circle
endpoint. In this manuscript, the spectrum of all self-adjoint extensions for a
general exceptional Laguerre-type differential expression is given in terms of
the Darboux transformations which relate the expression to the classical
Laguerre differential expression. The spectrum is extracted from an explicit
Weyl $m$-function, up to a sign.
  The construction relies primarily on two tools: boundary triples, which
parameterize the self-adjoint extensions and produce the Weyl $m$-functions,
and manipulations of Maya diagrams and partitions, which classify the seed
functions defining the relevant Darboux transforms. Several examples are
presented.
