In a large skillet over medium heat, add the 2 tablespoons butter and onions and cook slowly until soft but not brown, about 10 minutes. Transfer to a mixing bowl and set aside.
To the mixing bowl, add the beef, 2 tablespoons softened butter, salt, pepper, thyme, and egg. Beat vigorously with a wooden spoon until well mixed.
Form mixture into patties ¾ inch thick and cover and chill until ready to use.
When ready to cook, lightly roll patties in flour spread on a plate, shaking off excess.
In a skillet over medium-high heat, heat the 1 tablespoon butter and oil. When it begins to foam and subside, sauté the patties to desired temperature, about 2-3 minutes or more per side depending on doneness.
Transfer patties to a plate and keep warm while you work on the sauce:
Drain fat from skillet. Add liquid (stock, wine, etc) and let boil, scraping up bits from bottom of pan as it reduces. Cook down to almost a syrup consistency, then turn off heat, and add butter by ½ tablespoons. Swirl pan to melt butter, then stir in herbs.
Spoon sauce over hamburger patties and serve. Enjoy!
