Dice one onion, mince the garlic, slice the mushrooms, and cut the remaining two onions into rings.

In a large frying pan, heat a bit of butter over medium heat and sauté the diced onion and garlic until softened.

In a large bowl, combine the ground beef, eggs, breadcrumbs, Dijon mustard, ketchup, oregano, thyme, salt, and pepper. Add the cooked onion and garlic mixture, then mix gently until just combined.

In the same pan, add the onion rings and cook over medium heat, stirring occasionally, until caramelized. Remove from the pan and set aside.

Form the beef mixture into six patties. In a 12-inch cast iron skillet, sear the patties on both sides for 3 - 5 minutes until browned (they don’t need to be cooked through). Remove and set aside.

Add the mushrooms to the same skillet and cook until browned, stirring occasionally.

In a bowl or large measuring cup, whisk together the beef broth, tomato paste, Dijon mustard, honey, Worcestershire sauce, balsamic vinegar, oregano, and thyme.

Pour the sauce mixture into the skillet with the mushrooms. Bring to a simmer, then ladle out about ½ cup of the hot liquid into a small bowl. Whisk the flour into this liquid until smooth, then return it to the skillet. Stir until the gravy thickens. Stir in the heavy cream. Season with salt and pepper to taste.

Return the patties and caramelized onions to the skillet. Simmer gently for 15 - 20 minutes, or until the patties are cooked through.