High-frequency-peaked BL Lacs (HBLs) are the dominant population of extragalactic sources at TeV energies, with more than 50 detected by ground-based TeV gamma-ray observatories. Between 2019 and 2022, the VERITAS observatory has conducted a campaign to observe a flux-limited sample of 36 X-ray selected HBLs in order to produce the first unbiased survey of TeV emission from HBL blazars. The VERITAS HBL sample includes 21 known TeV sources and 15 blazars which do not yet have reported TeV emission. The goal of these observations is to find unbiased estimates of the TeV flux of HBLs, which are generally only reported in their flaring states, and to detect new TeV blazars which have not previously been observed. Analysis of 25 out of 36 blazars in the sample has been completed, and two have been detected with significance above 5$\sigma$. This work presents detailed results of three sources and compares them to previously published work, as well as presenting overall results of the survey sample.