Doppler shifts in chromospheric and transition-region lines during solar flares are often interpreted as chromospheric condensation or evaporation. However, alternative sources of Doppler-shifted emission have been suggested, such as filament eruptions, jets or chromospheric bubbles. We analyse high-cadence scans from SORCE/SOLSTICE, which provide one-minute resolution profiles of the transition-region Si iii (1206 Å, $\textrm{T} = 10^{4.6}\,\textrm{K}$ ) line. 11 X-, M-, and C-class events observed during these scans with clear impulsive phase Si iii enhancements were identified. By subtracting a quiet-Sun profile and fitting Gaussian profiles to the Si iii line, measurements of flare-induced Doppler shifts were made. After correcting for a systematic trend in these shifts with solar longitude, two of the 11 events were found to exhibit a significant Doppler shift, one with a $201.36\pm 21.94\;\textrm{km\,s}^{-1}$ redshift and the other with a $-39.75\pm 11.00\;\textrm{km\,s}^{-1}$ blueshift. Intriguingly, SDO/AIA 304 Å and 1600 Å imaging revealed a bright eruption coincident with the event that exhibited a blueshift, suggesting the shift may have resulted from the eruption rather than evaporation alone. Our results highlight Si iii as a useful diagnostic of flaring dynamics at a temperature that has received limited attention to date. Future comparisons of these observations with radiative hydrodynamic simulations, along with new observations from state-of-the-art spectrometers such as SOLAR-C/EUVST and MUSE, should clarify the mechanisms behind the observed shifts in this study.