  Tracing the emergence of the massive quiescent galaxy (QG) population
requires the build-up of reliable quenched samples. We present Hubble Space
Telescope WFC3/G141 grism spectra of 10 quiescent galaxy candidates selected at
$2.5<z<3.5$ in the COSMOS field. Spectroscopic confirmation for the whole
sample is obtained within 1-3 orbits based on the presence of strong spectral
breaks and Balmer absorption lines. Combining their spectra with optical to
near-IR photometry, star-forming solutions are formally rejected for the entire
sample. Broad spectral indices are consistent with the presence of young A-type
stars, which implies that the last major episode of star formation has taken
place no earlier than $\sim$300-800 Myr prior to observation. Marginalising
over three different slopes of the dust attenuation curve, we obtain short
mass-weighted ages and an average peak star formation rate of SFR$\sim10^3$
M$_{\odot}$ yr$^{-1}$ at $z_{formation}\sim3.5$. Despite mid- and far-IR data
are too shallow to determine the obscured SFR on a galaxy-by-galaxy basis, the
mean stack emission from 3GHz data constrains the level of residual obscured
SFR to be globally below 50 M$_{\odot}$ yr$^{-1}$, hence three times below the
scatter of the coeval main sequence. Alternatively, the very same radio
detection suggests a widespread radio-mode feedback by active galactic nuclei
(AGN) four times stronger than in z$\sim$1.8 massive QGs. This is accompanied
by a 30% fraction of X-ray luminous AGN with a black hole accretion rate per
unit SFR enhanced by a factor of $\sim30$ with respect to similarly massive QGs
at lower redshift. The average compact, high S\'ersic index morphologies of our
galaxies, coupled with their young mass-weighted ages, suggest that the
mechanisms responsible for the development of a spheroidal component might be
concomitant with (or preceding) those causing their quenching.
