In a bowl, combine the cream cheese and gorgonzola cheese together.
Shape beef in 8 (4-inch) patties. Try not to 'over work' the patties.
Spoon the cheese mixture evenly into the center of 4 of the patties. Top with the remaining 4 patties. Press edges together to seal in the cheese. Cover patties and chill in the refrigerator for at least 30 minutes.
When patties are done chilling, preheat grill to about 350 to 400 degrees (medium to medium-high heat).
Place mushroom on half the sheet of foil. Drizzle with 2 Tbsp melted butter and sprinkle with ¼ tsp salt and pepper to taste. Fold over foil to form a pouch and tightly seal edges.
Season patties with garlic salt and pepper. Grill patties on preheated grill, covered for about 5-6 minutes per side. At the same time, grill mushrooms in foil packet for 10 minutes, turning once.
Serve patties on toasted buns with mayonaise and dijon mustard along with the mushrooms.