Peel mushroom tops and remove stems. Toss mushrooms with 4 teaspoons olive oil, 1 teaspoon balsamic vinegar, and garlic. Slice zucchini and squash into ⅛-inch-thick disks and toss with mushrooms. Add salt and pepper to taste. Let marinate 20 minutes.
Peel and julienne the onion. Saute in a saucepan with ½ teaspoon olive oil until onions turn golden brown. Then turn heat down, and add Port wine and 1 teaspoon balsamic vinegar. Reduce until dry. Set aside to cool.
In 4 cups hot water, add pinch of salt and ½ teaspoon olive oil. Place mozzarella curd in the water until pliable, about 30 seconds. Knead mozzarella into a ball shape, removing all air pockets. Wrap in plastic and submerge in ice water. After chilling, slice the mozzarella into ⅛-inch-thick disks.
In a hot skillet, saute mushrooms, zucchini, and squash until cooked through. In a separate pan, reheat onions. Place a mushroom (to form a base for the dish) onto a roasting pan. In consecutive layers, add some of the charred onions, a few rounds of zucchini, mozzarella, and squash; repeat as you like. Place in oven at 350 degrees until cheese is melted, about 5 to 10 minutes.
Spread greens on a plate to form a bed for the mushroom. Sprinkle with garlic oil, scallion oil, balsamic vinegar, black pepper, and chervil. Serve portobello gateau in center of greens.