Set an oven rack 6 inches from the top element and heat the broiler to high. Completely remove the mushroom stems; thinly slice the stems. Set the mushroom caps stem side up on a heavy-duty rimmed baking sheet, drizzle with 1-½ Tbs. of the oil, and sprinkle with ½ tsp. salt and a few grinds of pepper. Broil until the mushrooms are brown and completely tender, 6 to 7 minutes.

Pulse the bread in a food processor until it forms uniform crumbs. In a large, heavy sauté pan, heat 2 Tbs. of the oil over medium-high heat. Add the breadcrumbs and ¼ tsp. salt, and cook, stirring constantly, until the breadcrumbs crisp and turn light brown, 4 to 5 minutes. Transfer to a plate and wipe the pan clean. Return the pan to high heat, add the remaining 1-½ Tbs. oil and the scallions, and sauté until browned and softened, about 2 minutes. Add the mushroom stems and ½ tsp. salt. Cook, stirring, until the stems soften and brown, about 3 minutes. Add the sherry and cook until almost evaporated, 30 to 60 seconds. Add the spinach and stir until it wilts, about 1 minute. Add the cream and 1/3 cup Parmigiano, reduce the heat to medium, and cook until the cream reduces slightly, about 2 minutes. Taste for salt and pepper; let cool.

Discard any liquid that may have accumulated in the mushroom caps. Mound the spinach mixture into the caps; top with the breadcrumbs (press them into the filling) and the remaining 1 Tbs. Parmigiano.

Heat the broiler to high. Broil the caps until the breadcrumbs brown a little more and the mushrooms heat through, 1 to 3 minutes. Squeeze the lemon half over the caps. Serve immediately.