The perfect transfer of single photons in quantum systems is crucial for advancing quantum information and quantum communication technologies. Chiral waveguide quantum electrodynamics (QED) setups serve as promising platforms in this context. In chiral onedimensional (1D) waveguides, photon emission becomes unidirectional due to the spin-orbit interaction of light. In principle, this would allow a qubit to emit a photon with unit probability towards a second qubit, increasing the probability of absorption by the latter. However, with a linear dispersion relation in the chiral waveguide, single-photon absorption is fundamentally limited to about 54% due to the constraints on the shape of the single-photon pulse governed by the timereversal symmetry of the Schrödinger equation. In the literature, these constraints on the pulse shape for efficient absorption have been overcome by using cavities, spectrally engineering the couplings or modeling time-dependent couplings. In this work, we explore an alternative approach, namely to engineer the dispersion relation of the chiral waveguide such that the single-photon pulse is reshaped, just by free propagation in the waveguide, to enhance its absorption. Specifically, we investigate a system of two identical qubits, which are chirally coupled to a 1D photonic waveguide. The first qubit emits an exponentially decaying pulse into the waveguide, yet the second qubit, waiting to complete the time-reverse process, desires an exponentially growing pulse from the waveguide. Without dispersion engineering, this mismatch in pulse shapes results in only 54% absorption in the second qubit. With dispersion engineering, we show it is possible to turn the exponentially decaying pulse into an exponentially increasing pulse, achieving 100% energy absorption in the Markov limit. We are extending our approach to the non Markov region where qubits can be much closer, and exploring experimentally feasible photonic platforms. Compared to existing techniques, our work offers a high-efficient small-footprint solutions for on-chip quantum communications.