In a small saucepan, cook the port over moderate heat until reduced to 2 tablespoons, about 15 minutes.
Heat a cast-iron griddle until very hot. Form the meat into four 4-by-1-inch patties without packing too tightly. Season generously with salt and pepper. Add the patties to the griddle, cover with a roasting pan and cook over moderately high heat for 4 minutes, until very crusty. Flip the patties and cook, covered, for 2 minutes longer; top with the Stilton and cook uncovered for 1 minute. Transfer the patties to a plate and sprinkle with the umami dust, let rest for 2 minutes and set on the buns. Drizzle with the reduced port, top with the buns and serve.