In skillet, heat 1 tablespoon oil over medium heat. Add tomatoes and cook 1 minute. Add mushrooms and gently incorporate with tomatoes. Add water and stir until water evaporates and mushrooms darken and become tender, about 4 minutes. Add thyme and salt, to taste. Set aside to cool.
To assemble, gently smash and spread half of each avocado over one slice of toast. Top each slice of toast with mushroom mixture.
Top each with Parmesan cheese and drizzle with olive oil before serving.