Cover cooked asparagus with cheese, stick it under the broiler until the top is golden brown and crisp around the edges, and you have a fabulous and elegant dish. You can prepare any of the variations using asparagus instead of the suggested alternative vegetable. Other cooked vegetables you can use: green or wax beans, artichoke hearts, fennel, eggplant, leeks, spinach, celery root, parsnips, zucchini, winter squash or sweet potatoes.
Adjust an oven rack to about 4 inches from the heat source; turn on the broiler. Grease a shallow medium baking pan or gratin dish with the butter.
Bring a large pot of salted water to a boil. Add the asparagus and cook until bright green and tender, about 5 minutes. Remove to an ice bath to stop the cooking then drain on a paper towel-lined plate.
Add the cheese to the food processor and pulse until large crumbs form. Add the parsley and pulse 3 or 4 more times until it is about the same size as the cheese crumbs. Tear the bread into pieces and add to the food processor. Pulse a few times, then let the machine run for a few seconds, until coarsely chopped.
Put the asparagus in the dish and top with the breadcrumb mixture. Broil for 3 to 5 minutes, until the cheese is melted and golden. Sprinkle with a few grinds pepper and the paprika if you like, and serve.