In a small bowl, mix together the mozzarella cheese and green onion, set aside.
Next, using your hands, mix together the veal, Worcestershire sauce, garlic, salt & pepper. Form into 8 equal-size patties and flatten.
Top four of the patties with the cheese mixture, then top each patty with another patty and crimp the edges together to keep the cheese in during grilling.
Combine the mayo, sage and mustard, set aside.
Place the mushrooms on a baking sheet lined with aluminum foil. Pour a small amount of olive oil in a bowl. Using a brush, coat both sides of the mushrooms with oil and sprinkle with salt and pepper.
Grill the mushrooms for 10 - 15 minutes, until tender. Grill the burgers to medium.
Place burgers and mushrooms on a sesame bun and serve with sauce.