We explore a transverse-field Ising model that exhibits both spontaneous symmetry breaking and eigenstate thermalization. Within its ferromagnetic phase, the exact eigenstates of the Hamiltonian of any large but finite-sized system are all Schrödinger cat states: superpositions of states with "up" and "down" spontaneous magnetization. This model exhibits two dynamical phase transitions within its ferromagnetic phase: In the lowest-temperature phase the magnetization can macroscopically oscillate between up and down. The relaxation of the magnetization is always overdamped in the remainder of the ferromagnetic phase, which is divided into phases where the system thermally activates itself over the barrier between the up and down states, and where it quantum tunnels.