Place the mushrooms in a large ceramic bowl. Combine the olive oil, 2 tablespoons lemon juice and sugar in a screw-top jar. Shake until well combined. Pour marinate over the mushrooms. Cover and stand for 30 minutes.
Cut the tomatoes into 5mm-thick slices. Peel and slice the avocado. Tear the mozzarella into large pieces. Arrange the tomato slices over the base of a large platter. Top with avocado, mozzarella and mushrooms. Sprinkle with basil.
Combine the extra virgin olive oil and lemon rind. Drizzle over the salad. Season to taste. Serve.