To make Teriyaki Spread: Whisk together all ingredients until blended. Refrigerate, covered. (Yield: 6 cups)
For each serving, to order: Brush 2 mushroom caps and 1 onion slice with olive oil; season with salt and pepper. Grill on gas or charcoal grill until soft and browned; remove from grill and brush with 2 teaspoons teriyaki glaze. Brush cut sides of 1 roll with 2 tablespoons Teriyaki Spread. Layer bottom of roll with 2 lettuce leaves, 4 tomato slices, onion and mushrooms; cover with top. Grill on griddle or in sandwich press until warm and lightly browned on both sides. Serve warm with 2 tablespoons Teriyaki Spread on side for dipping.