In a mixing bowl, gently combine beef with the salt and pepper. Divide the meat into 8 equal portions. Quickly form each portion into a flat patty, handling the meat as little as possible. Sprinkle 4 of the patties with two tablespoons of the gorgonzola cheese. Top with the remaining four patties of beef and pinch the sides together to form a seal, indent the center. Refrigerate for 20 minutes, then set out at room temperature until ready to grill.
PREPARE THE MUSHROOMS:
Heat the butter in a large skillet until melted. Add the shallot, garlic and thyme and sauté for 3 minutes or until the shallot is tender. Add all of the mushrooms and season with salt and pepper. Sauté over medium-high heat until the mushrooms begin to brown. Remove the pan from heat and add the brandy and reserved mushroom-soaking liquid. Continue to cook this mixture until all of the liquid has evaporated and the mushrooms are tender.
PREPARE THE ONIONS:
In a large skillet, heat the oil until hot but not smoking. Add the onions, cover and cook over medium-low heat until onions are very soft and golden brown, about 25 minutes. Raise the heat to medium-high and add the vinegar and brown sugar. Cook until the liquid has almost evaporated and a thick syrup has formed. Add the heavy cream, salt and pepper to taste.
Grill the burgers until desired amount of doneness. Place one burger on each bun and top with one tablespoon of the onion marmalade and a heaping spoonful of the sautéed mushrooms.