We employ a cascaded system approach to numerically simulate the interaction of photon pulses with a two-level scatterer in a chiral waveguide QED setup and show that the scattering of a two-photon pulse may lead to the predominant population of only two output wave packet modes in an entangled state, |2,0)-|0,2). In a complementary wave-packet basis, this is a product state of two orthogonal single-photon wave packets. Operating this process in reverse allows a perfect merging of distinguishable single-photon wave packets into a single-mode pulse carrying two identical photons.