In a well-organized kitchen, a brown sauce is always improved if meat is first sweated and its juices caramelized along with the aromatic vegetables used to prepare the flavor base. Early French recipes almost invariably called for lining the bottom of the pot with a slice of ham or veal as a preliminary step in sauce preparation.
Unsmoked cured ham (prosciutto) will also contribute enormously to the flavor and complexity of a sauce. Cut the ham (use the ends; they’re cheaper) into cubes and sweat it along with the aromatic vegetables used to make the flavor base.