Preheat the broiler. Mist a foil-lined broiler pan with cooking spray. Put the mushrooms gill-side down on the pan; broil 5 minutes.
Meanwhile, toss the spinach, breadcrumbs, mozzarella, chopped shallot, tomato, 2 tablespoons each pecorino and olive oil, 1/2 teaspoon salt, and pepper to taste in a medium bowl.
Remove the mushrooms from the broiler and turn them gill-side up on the pan. Season with salt and pepper. Mound the spinach filling on the mushrooms and top with the ricotta. Sprinkle with the remaining 1 tablespoon pecorino and broil until the filling is heated through and the ricotta is lightly browned, about 5 minutes.
Meanwhile, toss the celery, sliced shallot and the remaining 1 tablespoon olive oil in a bowl and season with salt and pepper. Top with shaved pecorino. Divide the mushrooms among plates and serve with the celery salad. Drizzle with olive oil.