Snap off the tough ends from the asparagus and peel the lower half of each stalk if they are thick. Bring a large saucepan of salted water to a boil. Add the asparagus and cook until just tender, 5 to 8 minutes, depending on thickness. Drain, refresh in cold water to set the color, and drain again. Pat dry with paper towels or a kitchen towel.

Preheat the oven to 450°F. Grease a small gratin dish with about 1 teaspoon of the butter. Arrange the asparagus spears, tips facing in the same direction, in the prepared dish. If using the prosciutto, you may divide the spears into 2 bundles, or leave them in a single one, and wrap in a thin slice of prosciutto. Or you can dice the prosciutto and sprinkle it over the asparagus. Break the eggs over the asparagus and then sprinkle with some pepper and the Parmesan cheese. (You won’t need salt, as the cheese and prosciutto are rather salty.) Cut the remaining butter into bits and use it to dot the eggs and asparagus evenly.

Place in the oven and bake until the whites are set and yolks are still runny, 8 to 10 minutes. Eat piping hot.