A valve bridge assembly is disclosed for use with an internal combustion engine. The valve bridge assembly may have a valve bridge with a center portion, opposing lateral extensions, a central cavity formed within the center portion, and a bore formed within each of the opposing lateral extensions. The valve bridge assembly may also have a lash adjuster disposed within the central cavity, and a sleeve insert disposed within each bore of the opposing lateral extensions. Each sleeve insert may be configured to receive a corresponding valve stem.