Multilayer coated blazed gratings with high groove density are the best candidates for use in high resolution EUV and soft x-ray spectroscopy. Theoretical analysis shows that such a grating can be potentially optimized for high dispersion and spectral resolution in a desired high diffraction order without significant loss of diffraction efficiency. In order to realize this potential, the grating fabrication process should provide a perfect triangular groove profile and an extremely smooth surface of the blazed facets. Here we report on recent progress achieved at the Advanced Light Source (ALS) in fabrication of high quality multilayer coated blazed gratings. The blazed gratings were fabricated using scanning beam interference lithography followed by wet anisotropic etching of silicon. A 200 nm period grating coated with a Mo/Si multilayer composed with 30 bi-layers demonstrated an absolute efficiency of 37.6percent in the 3rd diffraction order at 13.6 nm wavelength. The groove profile of the grating was thoroughly characterized with atomic force microscopy before and after the multilayer deposition. The obtained metrology data were used for simulation of the grating efficiency with the vector electromagnetic PCGrate-6.1 code. The simulations showed that smoothing of the grating profile during the multilayer deposition is the main reason for efficiency losses compared to the theoretical maximum. Investigation of the grating with cross-sectional transmission electron microscopy revealed a complex evolution of the groove profile in the course of the multilayer deposition. Impact of the shadowing and smoothing processes on growth of the multilayer on the surface of the sawtooth substrate is discussed.