The escape fraction of Lyman continuum photons fesc(LyC) is the last key unknown in our understanding of cosmic reionization. Directly estimating the escape fraction (fesc) of ionizing photons in the epoch of reionization (EoR) is impossible, due to the opacity of the intergalactic medium (IGM). However, a high fesc leaves clear imprints in the spectrum of a galaxy, due to reduced nebular line and continuum emission, which also leads to bluer UV continuum slopes. Here, we exploit the large archive of deep JWST/NIRSpec spectra from the DAWN JWST Archive to analyze over 1400 galaxies at 5<z_spec_<10 and constrain their fesc based on SED fitting enhanced with a picket fence model. We identify 71 high-confidence sources with significant fesc based on Bayes factor analysis strongly favouring fesc>0 over fesc=0 solutions. We compare the characteristics of this high-escape subset against both the parent sample and established diagnostics including UVbeta slope, O32, and SFR surface density. For the overall sample, we find that most sources have a low escape fraction (<1%), however, a small subset of sources seems to emit a large number of their ionizing photons into the IGM, such that the average fesc is found to be ~10%, as needed for galaxies to drive reionization. Although uncertainties remain regarding recent burstiness and the intrinsic stellar ionizing photon output at low metallicities, our results demonstrate the unique capability of JWST/NIRSpec to identify individual LyC leakers, measure average fesc and thus constrain the drivers of cosmic reionization.