Combine all ingredients (for mayo) in food processor and pulse (or blender).
Sandwich: Using a pastry brush, brush olive oil on the vegetable slices and the portobello mushrooms.
Season them with salt and pepper.
Place vegetables on a hot grill (or under broiler) and cook until they are tender.
Once vegetables are done brush sliced bread with olive oil and grill (or broil) on both sides.
To assemble, slice portobellos into 1/4-inch slices, spread both sides of the grilled bread with herbed mayonnaise and then top with 1 slice each of grilled vegetables and a quarter of the mushrooms, and finish off with lettuce and top with remaining piece of bread.