Divide the ground beef into 4 equal portions and shape them into patties about 1/2 inch thick. Season both sides generously with salt and pepper.

Set up a breading station with three shallow bowls: one with flour, one with the beaten egg, and one with breadcrumbs mixed with the dried thyme.

Dredge each patty in flour, dip into the egg, and then press firmly into the breadcrumb mixture to coat evenly.

Heat 1 tablespoon of butter and the olive oil in a large skillet over medium heat. Fry the patties for 4-5 minutes per side until the crust is golden brown and the meat is cooked through. Remove patties and set aside on a plate.

In the same skillet, add the remaining 2 tablespoons of butter. Add the sliced onions and mushrooms. Sauté for 10-12 minutes, stirring occasionally, until the onions are caramelized and soft.

Pour in the beef broth and scrape up any browned bits from the bottom of the pan. Let it simmer for 2 minutes to reduce slightly.

Stir in the heavy cream and simmer for another 2-3 minutes until the sauce thickens.

Return the patties to the pan for 1 minute to warm through, spooning the sauce over them, then serve immediately.