An arbitrary high-order technique for numerically solving hyperbolic systems, e.g. the Euler equations for gasdynamics, with curved domains is presented. The proposed method is coupled with a Discontinuous Galerking (DG) discretization in space and the one-step ADER method to ensure high accuracy in time. The approach presented here, namely the Recostruction Off-site Data (ROD) [1], is based on a constrained least square problem with restrictions imposed on the curved boundaries of the domain that enables to handle general boundary conditions. This easily allows to straighly impose different kinds of boundary conditions (inlet, outlet, slip-wall, and so on) without experiencing the well-known second-order degradation in the simulations with curved domains, discretized using linear meshes. Thanks to the flexibility of the least square problem, the proposed method has already been applied to several applications as a black-box. The goal of this work is that to couple it with an arbitrary high-order ADER-DG method to run challenging time-dependent simulations with curved domain without experiencing order-of-accuracy degradation due to the geometrical error. However, it is well-known that hyperbolic equations experience discontinuities occurring in the field even when starting from smooth initial data. The ROD method, as it was conceived, does not take into account this possibility and for this reason the simulation can fail due to unphysical oscillations. In this work, we use a MOOD approach to detect when unphysical oscillations occur and adapt the ROD polynomials used to impose the enhanced boundary conditions also where shocks appear.