To prepare the burgers: Add the ground beef, stuffing, egg, parsley, and seasonings to a medium bowl. Use your hands to combine and form into 4 equal sized patties.
Heat a large skillet over medium heat. Melt 1 tablespoons of butter in the skillet and add the turkey patties.
Cook for 6 minutes before flipping. After flipping, chop the remaining tablespoon of butter into 4 small pieces and top each burger with a piece of butter.
Continue cooking for 6 more minutes or until a thermometer reads 160 degrees.
To prepare the gravy: While the burgers are cooking, melt the butter in a small sauce pan. Whisk in the flour and cook for 1 minute.
Whisk in the chicken stock, stirring constantly, until mixture thickens and gravy is smooth. Turn off heat.
To serve: Heat a large skillet over medium heat and melt the butter on the skillet.
Place the buns, cut side down, on the skillet to toast.
Top each bun with a turkey burger, scoop of mashed potatoes, chopped chives, and hefty drizzle of gravy. Top with bun and serve.