The new generation of solar instruments provides better spectral, spatial, and temporal resolution for a better understanding of the physical processes that take place on the Sun. Multiple-component profiles are more commonly observed with these instruments. Particularly, the He I 10830 Å triplet presents such peculiar spectral profiles, which give information on the velocity and magnetic fine structure of the upper chromosphere. The purpose of this investigation is to describe a technique to efficiently fit the two blended components of the He I 10830 Å triplet, which are commonly observed when two atmospheric components are located within the same resolution element. The observations used in this study were taken on 2015 April 17 with the very fast spectroscopic mode of the GREGOR Infrared Spectrograph (GRIS) attached to the 1.5-m GREGOR solar telescope, located at the Observatorio del Teide, Tenerife, Spain. We apply a double-Lorentzian fitting technique using Levenberg-Marquardt least-squares minimization. This technique is very simple and much faster than inversion codes. Line-of-sight Doppler velocities can be inferred for a whole map of pixels within just a few minutes. Our results show sub- and supersonic downflow velocities of up to 32 km s-1 for the fast component in the vicinity of footpoints of filamentary structures. The slow component presents velocities close to rest.