Silicon is well known for its strong third-order optical nonlinearity, exhibiting efficient supercontinuum and four-wave mixing processes. A strong second-order effect that is naturally inhibited in silicon can also be observed, for example, by electrically breaking the inversion symmetry and quasi-phase matching the pump and the signal. To generate an efficient broadband second-harmonic signal, however, the most promising technique requires matching the group velocities of the pump and the signal. In this work, we utilize dispersion engineering of a silicon waveguide to achieve group velocity matching between the pump and the signal, along with an additional degree of freedom to broaden the second harmonic through the strong third-order nonlinearity. We demonstrate that the strong self-phase modulation and cross-phase modulation in silicon help broaden the second harmonic by 200 nm in the O-band. Furthermore, we show a waveguide design that can be used to generate a second-harmonic signal in the entire near-infrared region. Our work paves the way for various applications, such as efficient and broadband complementary-metal oxide semiconductor based on-chip frequency synthesizers, entangled photon pair generators, and optical parametric oscillators.