Melt butter in a saucepan set over medium-low heat. Once melted sprinkle in flour and stir to combine. Continue stirring for approximately three minutes, or until until the mixture turns a light brown.
Slowly dribble in beef broth while continuing to stir. Ensure it is fully incorporated before more is added so lumps don't form. Add in the sprig of thyme and slowly add the wine as well.
Bring the gravy to a simmer and let cook until thickened, appoximately three minutes.
*Using 1/2 cup of red wine will create a fairly strongly flavored wine gravy. If you want a more mild wine-flavored gravy, cut the red wine in half and add an equal amount of additional beef broth.