  Context. Blazars are the most numerous class of High Energy (HE; E about 50
MeV - few 100 GeV) and Very High Energy (VHE; E about 100 GeV - 10 TeV)
gamma-ray emitters. As of today, a measured spectroscopic redshift is available
for only about 50% of gamma-ray BL Lacs, mainly due to the difficulty of
measuring reliable redshifts from their nearly featureless, continuum-dominated
optical spectra. The knowledge of the redshift is fundamental for understanding
the emission from blazars, for population studies and also for indirect studies
of the extragalactic background light and searches for Lorentz invariance
violation and axion-like particles using blazars. Aims. This paper is the first
of a series of papers which aim to measure the redshift of a sample of blazars
likely to be detected with the upcoming Cherenkov Telescope Array (CTA), a
ground based gamma-ray observatory. Methods. Monte Carlo simulations were
performed to select those hard spectrum gamma-ray blazars detected with the
Fermi-LAT telescope still lacking redshift measurements but likely to be
detected by CTA in 30 hours of observing time or less. Optical observing
campaigns involving deep imaging and spectroscopic observations were organised
to efficiently constrain their redshifts. We performed deep medium to high
resolution spectroscopy of nineteen blazar optical counterparts using the ESI
spectrograph at Keck, the RSS spectrograph at the SALT telescope, and the
EFOSC2 spectrograph at the ESO NTT. We searched systematically for spectral
features and, when possible, we estimated the contribution of the host galaxy
to the total flux. Results. We measured eleven firm spectroscopic redshifts
with values ranging from 0.1116 to 0.482. one tentative redshift, three
redshift lower limits including one at z > 0.449 and another at z > 0.868.
There were four objects found to have featureless spectra.
