We modify the scalar Einstein-aether theory by breaking the Lorentz invariance of a gravitational theory coupled to a Galileon type scalar field. This is done by introducing a Lagrange multiplier term into the action, thus ensuring that the gradient of the scalar field is time-like, with unit norm. The theory can also be considered as an extension to the mimetic dark matter theory, by adding some derivative self interactions to the action, which keeps the equation of motion at most second order in time derivatives. The cosmological implications of the model are discussed in detail. In particular, for pressure-less baryonic matter, we show that the universe experiences a late time acceleration. The cosmological implications of a special coupling between the scalar field and the trace of the energy-momentum tensor are also explored.