We demonstrate how to systematically design wave scattering systems with simultaneous coherent perfect absorbing and lasing operation at multiple and prescribed frequencies. The approach is based on the recursive assembly of non-Hermitian emitter and absorber units into self-dual emitter-absorber trimers at different composition levels, exploiting the simple structure of the corresponding transfer matrices. In particular, lifting the restriction to parity-time-symmetric setups enables the realization of emitter and absorber action at distinct frequencies and provides flexibility with respect to the choice of realistic parameters. We further show how the same assembled scatterers can be rearranged to produce unidirectional and bidirectional transparency at the selected frequencies. With the design procedure being generically applicable to wave scattering in single-channel settings, we demonstrate it with concrete examples of photonic multilayer setups.