We examine the capabilities of a fast and simple method to infer line-of-sight (LOS) velocities from observations of the photospheric Si I 10 827 Å line. This spectral line is routinely observed together with the chromospheric He I 10 830 Å triplet as it helps to constrain the atmospheric parameters. We study the accuracy of bisector analysis and a line core fit of Si I 10 827 Å. We employ synthetic profiles starting from the Bifrost enhanced network simulation. The profiles are computed solving the radiative transfer equation, including non-local thermodynamic equilibrium effects on the determination of the atomic level populations of Si I. We found a good correlation between the inferred velocities from bisectors taken at different line profile intensities and the original simulation velocity at given optical depths. This good correlation means that we can associate bisectors taken at different line-profile percentages with atmospheric layers that linearly increase as we scan lower spectral line intensities. We also determined that a fit to the line-core intensity is robust and reliable, providing information about atmospheric layers that are above those accessible through bisectors. Therefore, by combining both methods on the Si I 10 827 Å line, we can seamlessly trace the quiet-Sun LOS velocity stratification from the deep photosphere to higher layers until around logτ = -3.5 in a fast and straightforward way. This method is ideal for generating quick-look reference images for future missions like the Daniel K. Inoue Solar Telescope and the European Solar Telescope, for example.