Mix cheese and green onions in medium bowl. Gently mix ground veal, Worcestershire sauce, salt and pepper in large bowl. Divide veal mixture into 4 equal portions. Divide 1 portion in half; form two 1/8-inch-thick patties. Place 1/4 cup cheese mixture atop 1 patty; top with second patty. Seal at edges to enclose cheese mixture. Repeat with remaining 3 portions.
Mix mayonnaise, mustard, lemon peel and sage in small bowl. (Patties and mayonnaise mixture can be made 2 hours ahead. Cover separately and refrigerate.) Preheat oven to 400°F. Place mushrooms on rimmed baking sheet. Combine 4 tablespoons oil and garlic. Brush over both sides of mushrooms. Sprinkle with salt and pepper. Roast mushrooms until tender, turning once, about 20 minutes.
Heat remaining 2 tablespoons oil in heavy large skillet over medium-high heat. Add burgers and cook until cheese melts, about 5 minutes per side.
Spread mayonnaise mixture over buns. Place burgers on bottom halves. Top with mushrooms, then with bun tops.
