Embedded boundary methods (EBM) are particularly useful for simulating flow past complex geometries, as they significantly simplify the mesh generation process. Moreover, in the case of moving or deforming complex geometries, these methods eliminate the need for remeshing. This method is of interest for applications studying the potential use of thin, porous trailing-edge brushes for noise reduction in transonic aircraft. Several EBMs have been developed with numerical methods such as adaptive wavelet collocation, finite differences, and finite volume methods, while research on similar methods using the finite element method (FEM) for compressible flows is sparse. In this work, an embedded boundary space-time (ST) finite element method (FEM) with stabilization and discontinuity capturing (DC) was developed using Brinkman penalization (BPM). Stabilized FEM with DC has been widely used for high Reynolds number compressible flows involving shock waves. The method produces promising results in transonic flow regimes involving shock waves, where the temperature is allowed to evolve freely within the penalized region while a no-slip condition is imposed on the embedded structure. In these cases, the pressure and velocity fields throughout the fluid domain, as well as the temperature field away from the penalized region, are accurately captured. DC allows the use of discontinuous masking functions rather than regularized masking functions, which allows a more accurate representation of the embedded structure. The penalty terms drastically reduce the time step size required for stability in explicit-time integration schemes, while space-time FEM allows the use of larger time step sizes, as it is an implicit time integration scheme.