We consider critical one dimensional quantum systems initially prepared in their groundstate and perturbed by a smooth noise coupled to the energy density. By using conformal field theory, we deduce a universal description of the out-of-equilibrium dynamics. In particular, the full time-dependent distribution of atwo-point chiral correlation function can be obtained from solving two coupled ordinary stochastic differential equations. In contrast with the general expectation of heating, we demonstrate that the system reaches a non-trivial and universal stationary state characterized by broad distributions. As an example, we analyse the local energy density: while its first moment diverges exponentially fast in time, the stationary distribution, which we derive analytically, is symmetric around a negative median and exhibits a fat tail with 3/2 decay exponent. We obtain a similar result for the entanglement entropy production associated to a given interval of size $\ell$. The corresponding stationary distribution has a 3/2 right tail for all $\ell$, and converges to a one-sided Levy stable for large $\ell$. Our results are benchmarked via analytical and numerical calculations for a chain of non-interacting spinless fermions with excellent agreement.