Parametric optical nonlinearities are usually weak and require both high optical field intensity and phase-matching. Micro/nanophotonics, with strong confinement of light in waveguides of nanometer-scale crosssections, can provide high field intensity, but is still in need of a solution for phase-matching across a broad bandwidth. In this article, we show that mode-coupling in slot waveguides can engineer the waveguide modal dispersion, and with proper choice of materials, can achieve on-chip broadband second-harmonic phase-matching. A phase-matching bandwidth in the range of 220 nm at mid-infrared can occur for a hetero-slot waveguide consisting of aluminum nitride (AlN) and silicon nitride (SiN). With a high-nonlinearity polymer as cladding material, about 1.76 W-1cm-2 of normalized conversion efficiency in second-harmonic-generation (SHG) and about 23 dB signal gain in degenerate optical parametric amplification (DOPA) can be achieved over a broad bandwidth. An asymmetric-slot waveguide configuration and a thermal tuning scheme are proposed to reduce the fabrication difficulty. This concept of broadband second-harmonic phase-matching can be extended to other nonlinear optical frequency mixing processes, thus expanding the scope of on-chip nonlinear optical applications.