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

HST grism spectroscopy of z~3 massive quiescent galaxies: Approaching the metamorphosis