Cut-element methods, such as CutFEM, where the computational mesh is kept fixed and the domain boundary is allowed to cut arbitrarily through the mesh, are especially well-suited for shape-optimization applications. In this context, one particular advantage with the method is that the shape gradient can be computed exactly in the discrete case using an expression only containing boundary terms as opposed to the case when body-fitted meshes and mesh deformations are employed. Here we combine the CutFEM method with a level-set representation of the boundary to optimize the shape of a so-called phase plug in a compression driver, the standard sound source for midrange acoustic horns, ubiquitous in public address systems. A particular modeling challenge here is that viscothermal boundary-layer losses cannot be ignored, due to narrow chambers and slits in the device. Fortunately, we can rely on a recently developed and computationally inexpensive model to accurately model these losses~\cite{BeBeNo18}. The main issue in the design of the compression driver phase plug is to avoid resonance and interference phenomena. The question is then what objective function to choose in order to promote such designs. Here we utilize a lumped low-frequency circuit model of the device to compute an ideal frequency response, the one that we would obtain if resonance and interference effects would not appear. For the optimization, we then choose a least-squares tracking objective function towards this ideal frequency response. The optimization algorithm was then able to successfully design a set of phase plugs whose final frequency