Silicon integration of mid-infrared (MIR) photonic devices promises to enable low-cost, compact sensing and detection capabilities that are compatible with existing silicon photonic and silicon electronic technologies. Heterogeneous integration by bonding III-V wafers to silicon waveguides has been employed previously to build integrated diode lasers for wavelengths from 1310 to 2010 nm. Recently, Fabry-Pérot Quantum Cascade Lasers integrated on silicon provided a 4800 nm light source for mid-infrared (MIR) silicon photonic applications. Distributed feedback (DFB) lasers are appealing for many high-sensitivity chemical spectroscopic sensing applications that require a single frequency, narrow-linewidth MIR source. While heterogeneously integrated 1550 nm DFB lasers have been demonstrated by introducing a shallow surface grating on a silicon waveguide within the active region, no mid-infrared DFB laser on silicon has been reported to date. Here we demonstrate quantum cascade DFB lasers heterogeneously integrated with silicon-on-nitride-on-insulator (SONOI) waveguides. These lasers emit over 200 mW of pulsed power at room temperature and operate up to 100˝C. Although the output is not single mode, the DFB grating nonetheless imposes wavelength selectivity with 22 nm of thermal tuning.