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 1-1/2 Tbs. of the oil, and sprinkle with 1/2 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 1/4 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 1-1/2 Tbs. oil and the scallions, and sauté until browned and softened, about 2 minutes. Add the mushroom stems and 1/2 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.
The mushrooms can be broiled and stuffed up to two days ahead and refrigerated, covered. When ready to serve, bring to room temperature and then broil as directed above until the breadcrumbs are browned and the mushrooms are heated through.