In a large skillet over medium heat, add 2 tbsp butter and cook the onions slowly for about 10 minutes (until very tender but not browned). Place in a mixing bowl.
Add the beef, soft butter, seasonings, and egg to the onions in the mixing bowl and beat vigorously with a wooden spoon to blend thoroughly. Correct seasoning. Form into patties 3/4 inch thick. Cover with waxed paper and refrigerate until ready to use.
Just before sauteing, roll the patties lightly in the flour. Shake off excess flour.
Add 1 tablespoon butter and oil to a pan over moderately high heat and when you see the butter foam begin to subside, saute the patties for 2 to 3 minutes on each side, depending on how you like your hamburgers (rare, medium or well done).
Arrange the patties on the serving platter and keep warm for a moment while finishing the sauce. Pour the fat out of the skillet.
Add the liquid (broth, wine or water) and boil it down rapidly, scraping up the coagulated pan juices, until it has reduced almost to a syrup. Off heat, swirl leftover butter into the sauce until it is absorbed.
Pour the sauce over the hamburgers and serve.
Notes
Save these beef patties in a storage container and refrigerate for up