Prepare the Au Jus: Melt the butter in a saucepan over medium-high heat. Whisk in the flour until there are no lumps. Whisk half of the beef stock into the roux, continuing to whisk until the sauce begins to thicken. Add in the Worcestershire sauce and the remaining stock. Bring to a boil, reduce the heat, and simmer for 5 minutes. Season to taste. Set aside. 

Butter one side of the Texas Toast and lightly brown it on a griddle or in a frying pan. Set aside.

Pan-fry the burgers until they are your desired doneness in the center. Top with two slices of cheese and let it just begin to melt. Remove the burgers from the skillet. 

Build the Dunker burger by placing one cheese-covered patty on top of lightly browned Texas Toast. (toasted part to the inside)  and place the other piece of Texas toast (toasted to the inside) on top of the cheese. 

Butter the outsides of the bread and place the burger on a skillet or griddle and cook “Grilled cheese style” until the bread is lightly toasted, and the cheese is melted. 

Cut in half diagonally and serve with warm au jus for dipping.