The spectral element method combines Gauss-Lobatto quadrature with high-order Lagrange shape functions that interpolate Gauss-Lobatto points. The result is a diagonal mass matrix that can be used to build extremely efficient explicit time integration methods. In an immersed finite element setting, the special quadrature necessary on cut elements breaks the diagonality of the spectral element method. Moreover, poorly cut elements can arbitrarily reduce the critical time step size. We leverage several ideas to obtain a stable, efficient, and accurate immersed spectral element method. Generalized eigenvalue stabilization combined with the finite cell method eliminates the critical time step size restrictions imposed by cut cells. Instead of lumping the consistent mass matrix, we show that a few Krylov subspace iterations on cut elements sufficiently reduce the residual to retain stability and accuracy. The number of iterations required does not increase with mesh refinement. The matrix-free evaluation of both the residual vector and the consistent mass of cut elements improves the algorithmic complexity and prevents poor parallel scaling due to memory throughput limitations. On cut elements, we distribute tensor-product quadrature points for which we compute custom weights (moment fitting). This talk provides a short overview of all the steps involved in constructing immersed spectral elements and demonstrates the accuracy and efficiency of this approach on a set of benchmark problems.