We study dynamically highly asymmetric binary mixtures comprised of small methyl tetrahydrofuran (MTHF) molecules and polystyrene. Combined use of dielectric spectroscopy, 2H nuclear magnetic resonance, incoherent quasielastic neutron scattering, and depolarized dynamic light scattering allows us to selectively probe the dynamics of the components in a broad dynamic range. It turns out that the mixtures exhibit two glass transitions in a wide concentration range although being fully miscible on a macroscopic scale. In between both glass transition temperatures, the dynamics of the small molecules show strong confinement effects, e.g., a crossover from Vogel-Fulcher to Arrhenius behavior of the time constants. Moreover, the dynamical behavior of small molecules close to the slow matrix is consistent with mode coupling theory predictions for a type-A glass transition, which was expected from recent theoretical and simulation studies in comparable systems.