We present a new kind of weighted essentially non-oscillatory (WENO) schemes to high-order continuous (CG) and discontinuous Galerkin (DG) discretizations of hyperbolic conservation laws. Unlike Runge-Kutta DG schemes that overwrite finite element solutions with WENO reconstructions, our approach uses a reconstruction-based smoothness sensor to control the amount of added numerical dissipation. The so-defined WENO approximation introduces low-order nonlinear diffusion in the vicinity of shocks, while attaining high order in regions where the exact solution is smooth. As such, our approach offers an attractive alternative to WENO-based slope limiters for DG schemes. The underlying reconstruction procedure performs Hermite interpolation on stencils consisting of a mesh cell and its neighbors. The amount of numerical dissipation depends on the relative differences between partial derivatives of reconstructed candidate polynomials and those of the consistent finite element approximation. All derivatives are taken into account by the employed smoothness sensor. To assess the accuracy of our WENO scheme, we derive a priori and a posteriori error estimates. Numerical experiments demonstrate the ability of our scheme to capture discontinuities sharply. For uniform meshes and smooth exact solutions, the experimentally observed rate of convergence is as high as p + 1.