Arrange a rack in the lower third of the oven and preheat to 400 degrees .
In a large skillet, heat 2 tablespoons olive oil over medium heat. Add the onion, celery, carrots, 1/2 teaspoon salt and 1/4 teaspoon pepper and cook until crisp-tender, about 5 minutes. Increase the heat to medium-high. Add the mushrooms and cook until tender, about 6 minutes. Sprinkle the flour on top; stir for 1 minute. Stir in the cream and simmer, stirring occasionally, until slightly thickened, about 3 minutes. Stir in the chicken, parsley and parmesan; season with salt and pepper.
Transfer the chicken mixture to an 8-inch-by-10-inch baking dish. Trim the bread slices to fit on top of the chicken mixture in an even layer. Brush with olive oil and sprinkle parsley and the remaining parmesan on top. Bake until crispy, about 15 minutes.