Employing the conformal welding technique, we find an exact expression for the Full Counting Statistics of energy transfers in a class of inhomogeneous nonequilibrium states of a (1+1)-dimensional unitary Conformal Field Theory. The expression involves the Schwarzian action of a complex field obtained by solving a Riemann-Hilbert type problem related to conformal welding of infinite cylinders. On the way, we obtain a formula for the extension of characters of unitary positive-energy representations of the Virasoro algebra to 1-parameter groups of circle diffeomorphisms and we develop technique, based on the analysis of certain classes of Fredholm operators, that allows to control the leading aymptotics of such extensions for small real part of the modular parameter τ .