Bifteck Hache a la lyonnaise

This take hamburgers to a whole new level.  This was adapted the "Mastering the Art of French Cooking":

Cook 3/4 cup minced onions in 2 tbsp butter until soft, but not browned, about 10 minutes.  Mix with 1 pound ground beef, 1 1/2 tbsp softened butter, 1 tsp salt, 1/8 tsp pepper, 1/8 tsp thyme and 1 egg.  Form into patties.  Roll lightly in flour and shake off excess.  Heat 1 tbsp butter and 1 tbsp olive oil.  Cook patties in batches, about 3 minutes each side for med rare.  Keep warm.  (At this point I added one chopped portabella mushroom and sauteed until soft.)  Add 1/2 cup beef stock and reduce until syrupy.  Remove from heat, and stir in 2 tbsp butter.  Yummy!