We present a finite element method for embedded boundary computations of compressible inviscid flows that are governed by the Euler equations. It belongs to the class of surrogate/approximate boundary algorithms and is based on shifting the location where boundary conditions are applied from the true boundary to a surrogate one; hence it is called the shifted boundary method or SBM for short. While SBM can be utilized to enforce a large variety of boundary conditions, we focus on the imposition of the normal component of flow velocity at an embedded boundary, which is typical in fluid-structure interaction applications. To this end, the method is composed of a finite element discretization of the underlying flow equation using piecewise linear elements, equipped with a variational multiscale stabilization (VMS) to improve linear stability and an entropy viscosity to capture strong discontinuities, with the shifted transmission condition enforced weakly by a Nitsche type method. Numerous numerical examples are presented to assess the numerical performance of the proposed method; and they involve both stationary and moving boundaries. In particular, because SBM distinguishes the normal vector to the true boundary from that to the surrogate one, we will demonstrate that it can be utilized to improve the accuracy of a conformal-grid computation at curved boundaries.