We are going to present a high-order accurate numerical method for simulating multiphase flows with phase change phenomena, specifically focusing on the Leidenfrost effect on soft substrates. Beside phase-change, fluid structure interaction (FSI) is employed to capture the deformation of the soft substrate under the influence of the vapor layer formed between the droplet and the hot solid surface. A key feature of such multiphase flow problems is the presence of discontinuities in the solution fields across the fluid interfaces. These discontinuities are e.g., jumps in the pressure field due to surface tension effects, or jumps in the velocity due evaporation. Numerically, such flows are challenging, especially for high-order methods which have a tendency to produce spurious oscillations or degenerate in accuracy in the vicinity of discontinuities (Gibbs phenomenon) therefore, the spatial discretization is based on the Extended Discontinuous Galerkin (XDG) method, which allows for high-order accurate approximations of the solution fields in the presence of discontinuities at interfaces. The presented method utilizes a Level-Set method to embed both interfaces, i.e., the fluid-fluid interface between liquid and vapor and the fluid-solid interface between vapor and solid, within a fixed Cartesian background mesh. In those computational cells that are cut by an interface, the standard discontinuous Galerkin element space is extended to be conformal with the interface geometry. This allows to accurately represent the discontinuities in the solution fields. In the presentation, we are going to present crucial aspects of the numerical method, as well as various numerical examples of Leidenfrost droplets on soft as wells as rigid substrates.