Mix and shape the patties: In a large mixing bowl, combine the ground beef, panko breadcrumbs, beaten egg, minced garlic, grated yellow onion, Worcestershire sauce, tomato paste, Dijon mustard, salt, and black pepper. Mix gently with clean hands just until combined—do not overwork the meat. Shape into 4 oval patties about 3/4-inch thick. Use your thumb to make a slight shallow depression in the center of each patty so they hold their shape while searing.

Sear the patties: Heat 1 tablespoon olive oil and 1 tablespoon butter in a large cast-iron skillet over medium-high heat. Add the patties and sear without moving for 4 to 5 minutes per side until deeply golden browned and crusted. Transfer the patties to a plate (they will finish cooking in the gravy later).

Sauté mushrooms and onions: Melt the remaining 1 tablespoon butter in the same skillet over medium heat, scraping up the flavorful browned fond from the bottom. Add the sliced onions and cremini mushrooms. Sauté for 6 to 7 minutes until the onions turn golden brown and caramelized, and the mushrooms are tender and fragrant. Stir in the minced garlic and dried thyme for 60 seconds.

Build the rich gravy: Sprinkle the 2 tablespoons of flour evenly over the sautéed onions and mushrooms. Stir and cook for 1 to 2 minutes to cook out the raw flour taste. Slowly pour in the beef broth while whisking continuously to prevent lumps. Whisk in the Worcestershire sauce and Dijon mustard. Bring the gravy to a gentle simmer, allowing it to thicken into a velvety, glossy sauce (about 2 to 3 minutes). Season with salt and pepper to taste.

Simmer patties in gravy: Nestlé the seared beef patties along with any resting juices back into the bubbling skillet. Spoon the rich mushroom gravy and onions generously over the tops of each patty. Reduce the heat to medium-low, cover partially, and simmer gently for 6 to 8 minutes until the patties are cooked through (internal temperature reaches 160°F / 71°C) and tender.

Garnish and serve: Remove skillet from heat and garnish generously with chopped fresh parsley. Serve hot, ladling generous spoonfuls of the caramelized onion and mushroom gravy over creamy garlic mashed potatoes and crisp green beans.