Combine the first six ingredients in a bowl. Moisten your hands with cold water and shape the meat into four thin patties. Set an equal amount of blue cheese in the centre of each patty. Fold the meat over the cheese and ensure it is complete encased. Now press the meat into 3/4" thick patties.
Preheat your barbecue or indoor grill to medium-high. Grill the burgers about 4 minutes per side, or until entirely cooked through and the centre of the burger reaches 160 degrees F on an instant read meat thermometer.
To make the sauce (it can be made ahead and reheated), place the wine in a small pot and reduce by half. Add the stock/flour mixture, bring to a simmer and cook until lightly thickened. Whisk in the mustard and season with salt and pepper.
To serve, plate the hamburger steaks and top with sauce.