  We show that excited-state quantum phase transitions (ESQPTs) in a system in
which the parity symmetry is broken can be used to engineer an energy-cat state
-- a Schr\"odinger cat state involving a quantum superposition of both
different positions and energies. By means of a generalization of the Rabi
model, we show that adding a parity-breaking term annihilates the ground-state
quantum phase transition between normal and superradiant phases, and induces
the formation of three excited-state phases, all of them identified by means of
an observable with two eigenvalues. In one of these phases, level crossings are
observed in the thermodynamic limit. These allow us to separate a wavefunction
in two parts: one, with lower energy, trapped within one region of the
spectrum, and a second one, with higher energy, trapped within another.
Finally, we show that a generalized microcanonical ensemble, including two
different average energies, is required to properly describe equilibrium states
in this situation. Our results illustrate yet another physical consequence of
ESQPTs.
